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\title[Nonlinear elliptic problems on Riemannian manifolds]
{Nonlinear elliptic problems on Riemannian manifolds and applications to Emden-Fowler type equations}

\author{Gabriele Bonanno, Giovanni Molica Bisci and Vicen\c{t}iu D.R\u{a}dulescu}
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\address[G. Bonanno]{Department of Science for Engineering and Architecture (Mathematics Section),
Engineering Faculty, University of Messina, 98166 - Messina, Italy.}
\email{bonanno@@unime.it}

\address[G. Molica Bisci]{Dipartimento MECMAT, University of Reggio Calabria,
 Via Graziella, Feo di Vito, 89124 Reggio Calabria, Italy.} \email{gmolica@@unirc.it}

\address[V. R\u{a}dulescu]{Institute of Mathematics ``Simion Stoilow" of the Romanian Academy, 014700 Bucharest, Romania. Department of Mathematics, University of Craiova,  200585 Craiova, Romania.} \email{vicentiu.radulescu@@imar.ro}

\keywords{Riemannian manifold, Emden-Fowler equation, critical eigenvalue problem, Cerami condition, multiple solutions.}
\thanks{{\em 2010 Mathematics Subject Classification.} 35J60, 58J05.}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{proposition}{Proposition}[section]
\newtheorem{corollary}{Corollary}[section]
\newtheorem{remark}{Remark}[section]
\newtheorem{Definition}{Definition}[section]
\newtheorem{example}{Example}[section]
\newtheorem{th}{Theorem}
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\newcommand{\erre}{\mathbb R}
\newcommand{\enne}{\mathbb N}
\def\essinf{\mathop {\rm ess\,inf}}
\def\esssup{\mathop {\rm ess\,sup}}
\def\RR{{\mathbb R}}
\def\N{{mathbb N}}
\def\essinf{\mathop {\rm ess\,inf}}
\def\n{\noindent}
\def\phi{\varphi}
\def\div{\mathop{\rm div}}
\def\meas{\mathop{\rm Vol}}
\def\det{\mathop{\rm det}}
\def\H{\mathop{H^1_0(\Omega)}}
\def\xseg{\bar{x}}
\def\useg{\bar{u}}
\def\d{\mathop{\rm dist}}
\def\R{{\mathbb R}}
\def\N{{\mathbb N}}
\def\weak{\rightharpoonup}
\def\ep{\varepsilon}
\def\half{{1\over2}}
\def\phi{\varphi}
\def\supp{{\rm supp}}
\def\di{\displaystyle}
\def\ri{\rightarrow}
\def\intom{\int_\Omega}
\begin{document}


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\begin{abstract}
The existence of one non-trivial solution for a nonlinear problem on compact $d$-dimensional ($d\geq 3$) Riemannian manifolds without boundary, is established. More precisely, a recent critical point result for
differentiable functionals is exploited, in order to prove the
existence of a determined open interval of positive eigenvalues for
which the considered problem admits at least one non-trivial weak solution. Moreover, as a consequence of our approach, a multiplicity result is presented requiring the validity of the Ambrosetti-Rabinowitz hypothesis. Successively, the Cerami compactness condition is studied in order to obtain a similar multiplicity theorem in superlinear cases. Finally, applications to Emden-Fowler type equations are presented.
\end{abstract}
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\section{Introduction}
\indent
The purpose of the present paper is to establish a new existence result associated with related energy estimates for elliptic problems defined on compact Riemannian manifolds.

Let $({\mathcal{M}},g)$ be a compact $d$-dimensional Riemannian manifold without boundary of dimension $d\geq 3$ and let $\Delta_{g}$ denote the  Laplace-Beltrami operator whose expression, in local coordinates,
is given by
$$
\Delta_g w:=g^{ij}\left(\frac{\partial^2w}{\partial x^i\partial x^j}-\Gamma^{k}_{ij}\frac{\partial w}{\partial x^k}\right),
$$
where $\Gamma^{k}_{ij}$ are the usual Christoffel's symbols.

We are interested in the existence
of one non-trivial weak solution of the following non-autonomous problem
$$-\Delta_{g} w+\alpha (\sigma) w= \lambda
K(\sigma)f(w), \eqno{(P_{\lambda})}$$
\noindent for every $\sigma\in {\mathcal{M}}$ and $w\in H_1^2({\mathcal{M}})$.


We assume that the mappings $\alpha, K:{\mathcal{M}}\to \erre$ satisfy
$$
\alpha,K\in \Lambda_+({\mathcal{M}}):=\left\{\beta\in L^{\infty}({\mathcal{M};\erre}):\essinf_{\sigma\in{\mathcal{M}}}\beta(\sigma)>0\right\},
$$
 \noindent $\lambda$ is a positive parameter, and the nonlinearity $f:\erre\to \erre$ is continuous and
$$
|f(t)|\leq a_1+a_2|t|^{q-1},\,\,\,\,\,\forall\, t\in\erre,\leqno({\rm h}_\infty)
$$
\noindent {for some non-negative constants $a_1,a_2$, and $q\in ]1,
2^{*}[$, where $2^{*}:=2d/(d-2).$}\par

 A remarkable case of problem $(P_{\lambda})$ is
$$-\Delta_{h}w+ s(1-s-d)w= \lambda
K(\sigma)f(w),\qquad \sigma\in {\mathbb{S}}^d,\, w\in H_1^2({\mathbb{S}}^d),\eqno(S_\lambda)
$$
 where ${\mathbb{S}}^d$ is the unit sphere in ${\erre}^{d+1}$, $h$ is the standard metric induced by the embedding ${\mathbb{S}}^d\hookrightarrow {\erre}^{d+1}$, $s$ is a constant such that $1-d<s<0$, and $\Delta_h$ denotes the Laplace-Beltrami operator on $({\mathbb{S}}^d,h)$.\par
 For completeness we  observe that the existence of a smooth positive solution for problem $({{S}}_\lambda)$, when $s=-d/2$ or $s=-d/2+1,$ and
$f(t)=|t|^{\frac{4}{d-2}}t,$
can be viewed as an affirmative answer to the Yamabe
problem \cite{yamabe} on ${\mathbb{S}}^d$ (see also the Nirenberg problem \cite{nire}); for these topics we refer to Aubin \cite{A-konyv}, Cotsiolis and Iliopoulos \cite{CI-BUll-Sci-math, CI-C.R.Acad}, Hebey \cite{Hebey}, Kazdan and Warner \cite{kazwar}, V\'azquez and  V\'eron \cite{VV-C.R.Paris}, and to the excellent survey by Lee and Parker \cite{leepar}. In these cases the right hand-side of problem $(S_\lambda)$ involves the critical Sobolev exponent. The Yamabe problem is often referred to in the literature on
elliptic type equations with critical Sobolev growth and in terms of PDEs equations can be formulated as follows:\par
\textit{For any smooth compact Riemannian manifold $({\mathcal{M}},g)$ of dimension $d\geq 3$, there exists $w\in C^{\infty}(\mathcal{M})$, $w>0$, and $\lambda\in \erre$ such that}
  $$-\Delta_{g} w+\frac{d-2}{4(d-1)}S_gw= \lambda
w^{\frac{d+2}{d-2}}, \eqno{(Y_{\lambda})}$$
\textit{where $S_g$ denotes the scalar curvature that, through the symmetric Ricci tensor $R_{ij}$, has the form $S_g=R_{ij}g^{ij}$.}\par
Geometrically, the goal of this celebrated problem is to
prove that, up to conformal changes of the metric, there always exists a metric of constant scalar
curvature. This was announced to be true by Yamabe \cite{yamabe} in 1960. Roughly eight years later,
Trudinger \cite{Trudinger} discovered a serious difficulty in the original Yamabe's proof. He repaired the proof when
the conformal class of the reference metric is nonpositive. Eight years later after Trudinger, Aubin \cite{Au} improved Yamabe's approach and reduced the problem to the proof of some
strict inequality involving some geometrical invariants of the manifold. Such an inequality was proved to be
true by Aubin in some cases, and then by Schoen \cite{S} in the remaining more difficult cases.
In particular, in his remarkable work, Schoen discovered the unexpected relevance of the
positive mass theorem. The Yamabe problem, whose origin goes back to the beginning of the
1960's, was solved something like twenty five years later.\par

Moreover, existence results for problem $(S_\lambda)$, can be used in order to study the existence of solutions for the following parameterized Emden-Fowler equation
 $$-\Delta u =
\lambda|x|^{s-2}K({x}/{|x|})f(|x|^{-s}u),\qquad x\in {\erre}
^{{d+1}}\setminus \{0\},\eqno(\mathfrak{F}_\lambda)$$
see, for instance, Section 4.\par
Problem $(P_{\lambda})$ has been studied for power-type nonlinearities, that is, provided that $f(t)=|t|^{p-1}t$, $p>1$ see Cotsiolis and Iliopoulos in \cite{CI-BUll-Sci-math, CI-C.R.Acad} for the case of the sphere, and V\'{a}zquez and V\'{e}ron \cite{VV-C.R.Paris} for a general compact manifold. In the aforementioned papers the authors obtained existence and multiplicity results for $(P_{\lambda})$ by using variational arguments.\par
More recently, in Krist\'aly and R\u adulescu \cite{KR}, the authors are interested on the existence of multiple solutions of problem $(P_{\lambda})$ in order to obtain solutions for parameterized Emden-Fowler equation $(\mathfrak{F}_\lambda)$ considering nonlinear terms of sublinear type at infinity.\par
In particular, in \cite[Theorem 1.1]{KR}, for $\lambda$ sufficiently large, the existence of two nontrivial solutions for problem $(P_{\lambda})$ has been successfully obtained through a careful analysis of the standard mountain pass geometry.\par
Furthermore, in Krist\'aly, R\u adulescu and Varga \cite[Theorem 9.4, p. 222]{KRV}, the existence of an open interval of positive parameters for which problem $(P_{\lambda})$ admits two distinct nontrivial solutions is established by using an abstract three critical points theorem contained in Bonanno \cite{BONANNO}.\par
Moreover in \cite{BOMORa}, through a novel approach developed by Bonanno and Molica Bisci \cite{BOMO},  the existence of a well localized open interval of positive parameters for which problem $(P_{\lambda})$ admits at least three non-trivial solutions has been studied.\par
Finally, Krist\'aly in \cite{K0}, proved a bifurcation result for a perturbed sublinear elliptic problem $(P_{\lambda,\mu})$ defined on $\mathcal{M}$. We just observe that for $\mu=0$ the cited problem coincides with $(P_{\lambda})$ and, in particular, from \cite[Theorem 1.1]{K0}, it follows that if the nonlinearity $f$ belongs to the set
$$
{\mathcal{F}}:=\left\{f\in C({\erre}_+;{\erre}_+)\setminus \{0\}:\lim_{t\rightarrow 0^+}\frac{f(t)}{t}=\lim_{t\rightarrow \infty}\frac{f(t)}{t}=0\right\},
$$
for $\lambda$ sufficiently small, then problem $(P_{\lambda})$ admits only the identically zero solution. For completeness, we just mention that, by using similar variational arguments, the existence of multiple solutions for non-homogeneous Neumann problem on Riemannian manifolds with boundary have been studied by Krist\'{a}ly, Papageorgiou and Varga in \cite{KPV2}.\par
The main result of this paper (Theorem \ref{TT1}) ensures the existence of precise values of parameters $\lambda$ for which ($P_\lambda$)
admits at least one non-trivial solution. A special case is also pointed out (Corollary \ref{CC2})
and a meaningful consequence for a suitable class of functions with a certain asymptotic behaviour at the origin is presented; see, for more details, Theorem \ref{ultimo}.\par
 We observe that when the nonlinear term is sublinear at infinity,
then the corresponding energy functional is coercive, hence
the existence of one solution (possibly zero) is ensured
from the direct methods theorem. It is worth noticing that, in our cases,
the potential may be also not coercive; see, for completeness, Example \ref{Ex1} and Remark \ref{remcoercivity}.
On the contrary, if the potential is coercive, our results ensure the existence of at least one non-trivial solution; see Remark \ref{direct}.\par

A basic tool used in the proofs is a recent critical point theorem obtained by Bonanno in \cite[Theorem 5.1]{Bonanno} for functionals of the form $J_\lambda:=\Phi-\lambda\Psi$, where $\lambda$ is a positive parameter; see Theorem \ref{T2} below.\par
We state in what follows a
special case of our results, which establishes the existence of a nontrivial solution in case of lower perturbations (small values of the parameter). The following theorem also shows that the energies of the corresponding solutions become smaller and smaller as the parameter converges to zero.

\begin{theorem}\label{introSfera}
Let $d\geq 3$. Set $\alpha,K\in C^\infty({\mathbb{S}}^d)$ be two positive maps and $f:\erre\to\erre$ be a continuous function such that
$$
\displaystyle\sup_{t\in\erre}\left(\frac{|f(t)|}{1+|t|^{q-1}}\right)<+\infty,
$$
for some $q\in ]1,2^*[$. Assume that
$$
\lim_{t\rightarrow0^+} \frac{f(t)}{t}=+\infty.
$$
 \noindent Then, there exists
$\lambda^{\star}>0$ such that for every $\lambda\in
]0,\lambda^\star[$, the following problem
$$-\Delta_{h} w+\alpha (\sigma) w= \lambda
K(\sigma)f(\omega),\qquad \sigma\in {\mathbb{S}}^d,\ w\in H_1^2({\mathbb{S}}^d),\eqno{(S_\lambda^{\alpha})}$$
admits at least one non-trivial weak solution $w_{0,\lambda}\in H_1^2({\mathbb{S}}^d)$.
Finally, $\|w_{0,\lambda}\|_{{H_1^2}}\rightarrow 0$ as
$\lambda\rightarrow 0^+$ and the map
$$
\lambda\mapsto \frac{1}{2}\displaystyle{\left(\int_{{\mathbb{S}}^d} |\nabla w_{0,\lambda}(\sigma)|^2 d\sigma_h+\int_{{\mathbb{S}}^d}w_{0,\lambda}(\sigma)^2 d\sigma_h\right)}-\lambda\int_{{\mathbb{S}}^d} K(\sigma)\left(\int_0^{w_{0,\lambda}(\sigma)}f(t)dt\right)d\sigma_h,
$$
is negative and strictly decreasing in $]0,\lambda^{\star}[$.
\end{theorem}
 \indent An explicit estimation of the parameter $\lambda^{\star}$ that appears in the above result is given in Remark \ref{maximalinterval}. For an exhaustive overview on elliptic equations with critical exponent defined on Riemannian manifolds we refer to \cite[Chapter 11]{KRV} as well as the exhaustive lecture notes \cite{noteH} and the references therein.\par
Finally, in Theorem \ref{CAR}, we prove that,
adding to hypotheses of Theorem \ref{ultimo} the classical $(\rm AR)$ Ambrosetti and Rabinowitz condition,  a second non-trivial solution is achieved; see also Corollary \ref{cork}.

Successively, in Theorem \ref{TheoremCerami} we establish a multiplicity result (similar to Theorem \ref{CAR}) without assuming $(\rm AR)$ condition. In this setting the Euler-Lagrange functional $J_\lambda$ may possess unbounded Palais-Smale (briefly $(\rm PS)$) sequences. The key point in our proof is that, although $J_\lambda$ possesses unbounded $(\rm PS)$ sequences, under the assumptions of Theorem \ref{TheoremCerami}, the functional $J_\lambda$ satisfies the Cerami condition.
Our result is achieved through a local condition, namely (\ref{Gigio}), near infinity, previously adopted by Liu in \cite{Liu} studying the existence of solutions for superlinear $p$-Laplacian Dirichlet problems on bounded Euclidean domains; see also the work of Jeanjean \cite{J}.\par


\indent
The paper is arranged as follows: in Section 2, we recall some basic definitions and our main tool, while Section 3 is devoted to our main results. In the last Section we find existence results for singular elliptic problems of Emden-Fowler type as an application of our theoretical approach. We cite the recent monograph by Krist\'aly, R\u adulescu and Varga \cite{KRV} as general reference on this subject.

\section{Preliminaries}

We start this section with a short list of notions in Riemmanian geometry. We refer to Aubin \cite{A-konyv} and Hebey \cite{Hebey} for detailed derivations of the geometric quantities, their motivation and further applications; see also the work \cite{aubin}. Let $({\mathcal{M}},g)$ be a smooth compact $d$-dimensional ($d\geq 3$) Riemannian manifold without
boundary and let $g_{ij}$ be the components of the metric $g$. As usual, we denote by $C^{\infty}({\mathcal{M}})$ the space of smooth functions defined on ${\mathcal{M}}$.
Let $\alpha\in \Lambda_+({\mathcal{M}})$ and set $\|\alpha\|_\infty:=\displaystyle\esssup_{\sigma\in {\mathcal{M}}} \alpha(\sigma)$.\par For every $w\in C^\infty ({\mathcal{M}}),$ we denote
$$\|w\|_{H_\alpha^2}^2:=\int_{\mathcal{M}} |\nabla w(\sigma)|^2 d\sigma_g+\int_{\mathcal{M}}\alpha(\sigma)w(\sigma)^2 d\sigma_g,$$ where
$\nabla w$ is the covariant derivative of $w,$ and
$d\sigma_g$ is the Riemannian measure. In local coordinates $(x^1,\ldots ,x^d)$, the components of $\nabla w$ are given by
$$
(\nabla^2 w)_{ij}=\frac{\partial^2w}{\partial x^i\partial x^j}-\Gamma^{k}_{ij}\frac{\partial w}{\partial x^k},
$$
where
$$
\Gamma^{k}_{ij}:=\frac{1}{2}\left(\frac{\partial g_{lj}}{\partial x^i}+\frac{\partial g_{li}}{\partial x^j}-\frac{\partial g_{ij}}{\partial x^k}\right)g^{lk},
$$
are the usual Christoffel's symbols and $g^{lk}$ are the elements of the inverse matrix of $g$. Here, and in the sequel, the Einstein's summation
convention is adopted. Moreover, the measure element $d\sigma_g$ assume the form $d\sigma_g=\sqrt{\det g}\,dx$, where $dx$ stands for the Lebesgue's volume element of ${\erre}^d$. Hence, let
$$
\textrm{Vol}_g({\mathcal{M}}):=\int_{{\mathcal{M}}}d\sigma_g.
$$
\noindent
In particular, if $({\mathcal{M}},g)=({\mathbb{S}}^d,h)$, where ${\mathbb{S}}^d$ is the unit sphere in ${\erre}^{d+1}$ and $h$ is the standard metric induced by the embedding ${\mathbb{S}}^d\hookrightarrow {\erre}^{d+1}$, we set
$$
\omega_d:=\textrm{Vol}_h({\mathbb{S}}^d):=\int_{{\mathbb{S}}^d}d\sigma_h.
$$
The Sobolev space
$H_\alpha^2({\mathcal{M}})$ is defined as the completion of $C^\infty({\mathcal{M}})$ with
respect to the norm $\|\cdot\|_{H_\alpha^2}.$ Then
$H_\alpha^2({\mathcal{M}})$ is a Hilbert space endowed with the inner product
$$\langle w_1,w_2\rangle_{H_\alpha^2}=\int_{\mathcal{M}}\langle
\nabla w_1(\sigma),\nabla w_2(\sigma) \rangle_g
d\sigma_g+\int_{\mathcal{M}}\alpha(\sigma)\langle w_1(\sigma),w_2(\sigma) \rangle_g
d\sigma_g,$$
for every $w_1,w_2\in H_\alpha ^2({\mathcal{M}})$,
where $\langle \cdot,\cdot \rangle_g$ is the
inner product on covariant tensor fields associated to $g.$ Since
$\alpha$ is positive, the norm $\|\cdot \|_{H_\alpha^2}$ is
equivalent with the standard norm
$$\|w\|_{H_1^2}:=\displaystyle{\left(\int_{\mathcal{M}} |\nabla w(\sigma)|^2 d\sigma_g+\int_{\mathcal{M}}w(\sigma)^2 d\sigma_g\right)}^{1/2}.$$
 \noindent Moreover, if $w\in H_\alpha^2({\mathcal{M}})$, the following inequalities hold
\begin{equation}\label{equivalance}
\small
  \min\{1,{\essinf_{\sigma\in{{\mathcal{M}}}}\alpha(\sigma)^{1/2}}\}\|w\|_{H_1^2}\leq
\|w\|_{H_\alpha^2} \leq
\max\{1,{\|\alpha\|_\infty^{1/2}}\}\|w\|_{H_1^2}.
\end{equation}
\indent By the Rellich-Kondrachov theorem for compact manifolds without boundary we have
$$
H_\alpha ^2({\mathcal{M}})\hookrightarrow L^q({\mathcal{M}}),
$$
for every $q\in [1,2d/(d-2)].$ In particular, the embedding is compact whenever $q\in [1,2d/(d-2))$.
\noindent Hence, there exists a positive constant $S_q$ such that
\begin{equation}\label{ineq}
\|w\|_{L^q(\mathcal{M})}\leq S_q \|w\|_{H_\alpha^2}, \qquad\forall\; w\in H_\alpha^2({\mathcal{M}}),
\end{equation}
where the norm of the Lebesgue spaces $L^q(\mathcal{M})$ are denoted by $\|\cdot\|_{L^q(\mathcal{M})}$ for all $q\in [1,\infty[$.\par
\indent If $K\in \Lambda_+({\mathcal{M}})$, we recall that a function $w\in H_1^2({\mathcal{M}})$ is a {\it{weak solution}} of problem $(P_{\lambda})$ if
$$
\int_{\mathcal{M}} \langle
\nabla w(\sigma),\nabla v(\sigma) \rangle_g
d\sigma_g+\int_{\mathcal{M}}\alpha(\sigma)\langle w(\sigma) ,v(\sigma) \rangle_g
d\sigma_g=\lambda\int_{\mathcal{M}} K(\sigma)f(w(\sigma))v(\sigma)d\sigma_g,\quad
$$
\noindent for every $v\in H_1^2({\mathcal{M}}).$\par

Let $X$ be a real Banach space. We say that a continuously G\^ateaux differentiable functional $J:X\rightarrow \erre$ verifies the {\it Palais-Smale condition}
(in short $(\rm PS)$-condition) if any sequence $\left\{u_n \right\}$
such that
\begin{itemize}
\item[$({\textrm{j}}_1)$] $\{J(u_n)\}$ is bounded,
\item[$({\textrm{j}}_2)$] $\displaystyle\lim_{n \to \infty}\|J'(u_n)\|_{X^*}=0$,
\end{itemize}
has a convergent subsequence.

\indent For an exhaustive treatment of these topics we refer to
\cite{Motreanu-Radulescu,Rabinowitz} and the references
therein. Now, let $\Phi, \Psi:X \to \erre$ be two continuously G\^ateaux
differentiable functions. Set $$J=\Phi-\Psi,$$
and fix $r_1, r_2 \in
[-\infty,+\infty]$, with $r_1<r_2$; we say that function $J$
verifies the {\it Palais-Smale condition cut off lower at $r_1$ and
upper at $r_2$} (in short $^{[r_1]}({\rm PS})^{[r_2]}$-condition) if any
sequence $\left\{u_n \right\}$ such that $({\textrm{j}}_1)$, $({\textrm{j}}_2)$ hold and
\begin{itemize}
\item[$({\textrm{j}}_3)$] $r_1<\Phi(u_n)<r_2$,\,\, $\forall\, n\in \enne$,
\end{itemize}
has a convergent subsequence.

\indent Clearly, if $r_1=-\infty$ and $r_2=+\infty$ it coincides
with the classical $({\rm PS})$-condition. Moreover, if $r_1=-\infty$ and
$r_2\in \erre$ it is denoted by $({\rm PS})^{[r_2]}$, while if $r_1\in
\erre$ and $r_2=+\infty$ it is denoted by $^{[r_1]}({\rm PS})$. Clearly, if
$J$ satisfies $^{[r_1]}({\rm PS})^{[r_2]}$-condition, then it
satisfies $^{[\rho_1]}({\rm PS})^{[\rho_2]}$-condition for all
$\rho_1,\rho_2 \in [-\infty,+\infty]$ such that $r_1\le
\rho_1<\rho_2\le r_2$.

  In particular, we deduce that if $J$ satisfies
the classical $({\rm PS})$-condition, then it satisfies
$^{[\rho_1]}({\rm PS})^{[\rho_2]}$-condition for all $\rho_1,\rho_2 \in
[-\infty,+\infty]$ with $\rho_1<\rho_2$.
Set
\begin{equation} \label{beta}
\beta(r_1,r_2):=\inf_{v \in \Phi^{-1}(]r_1,r_2[)} \frac{\displaystyle\sup_{u \in \Phi^{-1}(]r_1,r_2[)} \Psi(u)-\Psi(v)}{r_2-\Phi(v)},
\end{equation}
\noindent and
\begin{equation} \label{rho2}
\rho_2(r_1,r_2):=\sup_{v \in \Phi^{-1}(]r_1,r_2[)} \frac{\Psi(v)-\displaystyle\sup_{u \in \Phi^{-1}(]-\infty,r_1])} \Psi(u)}{\Phi(v)-r_1},
\end{equation}

\noindent for all $r_1, r_2 \in \erre$, with $r_1<r_2$.\par
Now, for a fixed $\lambda>0$, the function
$w_\lambda(\sigma)=c\in {\erre}\setminus\{0\} $, is a solution of $(P_\lambda)$
if and only if the mapping $\sigma\mapsto
{\lambda K(\sigma)}/{\alpha(\sigma)}$ is constant. In this case, nontrivial constant
solutions of $(P_\lambda)$ appear as fixed points of the function
$t\mapsto c_\lambda f(t),$ where $c_\lambda$ denotes the constant value ${\lambda K(\sigma)}/{\alpha(\sigma)}$.\par
A crucial role in the existence proof of one non-trivial (and non-constant) weak solution
for our problem is played by the following version of an
abstract local minimum theorem obtained in ~\cite[Theorem
5.1]{Bonanno}, which we recall  here for convenience.

\begin{theorem} \label{T2}
Let $X$ be a real Banach space and let $\Phi, \Psi :X \to \erre$ be two continuously
G\^ateaux differentiable functions.
Assume that there are $r_1$, $r_2 \in \erre$, with $r_1<r_2$, such
that
$$\beta(r_1,r_2)<\rho_2(r_1,r_2),$$
where $\beta$ and $\rho_2$ are given by \eqref{beta} and
\eqref{rho2}, and for each $$\lambda \in
\Big]\dfrac{1}{\rho_2(r_1,r_2)}, \dfrac{1}{\beta(r_1,r_2)} \Big[,$$
the function $J_{\lambda}:=\Phi-\lambda\Psi$ satisfies
$^{[r_1]}({\rm PS})^{[r_2]}$-condition.\par
 \noindent Then, for each $\lambda \in \Big]\dfrac{1}{\rho_2(r_1,r_2)},
\dfrac{1}{\beta(r_1,r_2)} \Big[$ there is $u_{0,\lambda} \in
\Phi^{-1}(]r_1,r_2[)$ such that $J_{\lambda}(u_{0,\lambda})\le
J_{\lambda}(u)$ for all $u \in \Phi^{-1}(]r_1,r_2[)$ and
$J_{\lambda}'(u_{0,\lambda})=0$.
\end{theorem}


\begin{remark}\label{R3}
{\rm We explicitly observe that Theorem \ref{T2} guarantees the
existence of an open interval of parameters such that for any

$$\lambda \in
\Big]\dfrac{1}{\rho_2(r_1,r_2)}, \dfrac{1}{\beta(r_1,r_2)} \Big[,$$

 \noindent the functional
$J_\lambda$ has a minimizer $u_{0,\lambda}$ with respect to the open
sublevel $\Phi^{-1}(]r_1,r_2[)$, that is,
$$
J_{\lambda}(u_{0,\lambda})\le J_{\lambda}(u),\quad\forall\
u\in\Phi^{-1}(]r_1,r_2[).
$$
Since $\Phi^{-1}(]-\infty,r[)\subset X$ is open,  it follows
that $u_{0,\lambda}$ must be a local minimizer of $J_\lambda$, hence
it is a critical point of $J_\lambda$. Therefore,
Theorem \ref{T2} may be considered also as a localization principle
of critical points of $J_\lambda$ that belong to the sublevel
$\Phi^{-1}(]r_1,r_2[)$.}
\end{remark}

\section{Main Results}
\indent For every two nonnegative constants $\gamma,\delta$, with $\gamma\neq\delta$, we set

\begin{equation}\label{g}
a_\gamma(\delta):= \frac{\displaystyle A(\gamma)-qF(\delta)\|K\|_{L^1(\mathcal{M})}}{\|\alpha\|_{L^1(\mathcal{M})}(\gamma^2-\delta^2)q},
\end{equation}
\noindent where
$$
A(\gamma):=(q\|\alpha\|_{L^1(\mathcal{M})}^{1/2}\gamma S_1a_1+\|\alpha\|_{L^1(\mathcal{M})}^{q/2}\gamma^{q}{S_q^qa_2})\|K\|_{\infty},
$$
and
$$F(\xi):=\int_0^{\xi}f(t)dt,$$
\noindent for every $\xi\in\erre$.\par
With the above notations we are able to prove the following result.

\begin{theorem}\label{TT1}
Let $f:\erre\to\erre$ be a continuous function such that condition $({\rm h}_\infty)$ holds and assume that there exist three real
constants $\gamma_1,\gamma_2$ and $\delta$, with $0\leq \gamma_1 <
\delta< \gamma_2$, such that
\begin{equation}\label{EE1}
a_{\gamma_2}(\delta)<a_{\gamma_1}(\delta).
\end{equation}
\noindent Then, for each parameter $\lambda$ belonging to
$$
\Lambda:=\left]\displaystyle \frac{1}{2a_{\gamma_1}(\delta)},\frac{1}{2a_{\gamma_2}(\delta)}\right[,
$$
the problem $(P_{\lambda})$ admits at least one weak solution $w_{0,\lambda}\in H_1^2({\mathcal{M}})$, such that $$\frac{\|\alpha\|_{L^1(\mathcal{M})}^{1/2}\gamma_{1}}{\max\{1,{\|\alpha\|_\infty^{1/2}}\}}<\|w_{0,\lambda}\|_{{H_1^2}}\leq \displaystyle\frac{\|\alpha\|_{L^1(\mathcal{M})}^{1/2}\gamma_2}{\min\{1,{\displaystyle\essinf_{\sigma\in{{\mathcal{M}}}}\alpha(\sigma)^{1/2}}\}}.$$
\end{theorem}

\begin{pf}
 Our aim is to apply Theorem \ref{TT1}. Hence, let $X:=H_1 ^2({\mathcal{M}})$ and consider the
functionals $\Phi,\Psi:X\to\erre$ defined by
$$
\Phi(w):={\frac{\|w\|_{H_\alpha^2}^2}{2}},\,\,\,\,
\displaystyle \Psi(w):=\int_{\mathcal{M}} K(\sigma)F(w(\sigma))d\sigma_g,\qquad\mbox{for all}\ w\in X.
$$
\noindent Clearly $\Phi:X\to\erre$ is a coercive, continuously G\^{a}teaux differentiable. On the other
hand, $\Psi$ is
well-defined and continuously G\^{a}teaux differentiable. Moreover, we have
$$
\Phi'(w)(v)=\int_{\mathcal{M}} \langle
\nabla w(\sigma),\nabla v(\sigma) \rangle_g
d\sigma_g+\int_{\mathcal{M}} \alpha(\sigma)\langle w(\sigma),v(\sigma)\rangle_g d\sigma_g,$$
\noindent and
$$
\Psi'(w)(v)=\int_{\mathcal{M}} K(\sigma)f(w(\sigma))v(\sigma)d\sigma_g,
$$
for every $w,v\in X$. Fix $\lambda>0$. A critical point of the functional $J_\lambda:=\Phi-\lambda\Psi$ is a
function $w\in X$ such that
$$
\Phi'(w)(v)-\lambda\Psi'(w)(v)=0,
$$
for every $v\in X$. Hence, the critical points of the functional $J_\lambda$ are the weak solutions of problem $(P_\lambda)$. Moreover, $\Phi (0_X)=\Psi (0_X)=0.$
 \noindent Since condition $({\rm h}_\infty)$ holds, we have
\begin{equation}\label{jj}
F(\xi)\leq a_1|\xi|+a_2\frac{|\xi|^{q}}{q},
\end{equation}
for every $\xi\in \erre$. Now, taking into account (\ref{jj}), it follows that
$$
\Psi(w)=\int_{\mathcal{M}} K(\sigma)F(w(\sigma))d\sigma_g\leq\|K\|_{\infty}\left( a_1\|w\|_{L^1(\mathcal{M})}+\frac{a_2}{q}\|w\|_{L^q(\mathcal{M})}^q\right).
$$
Then, for every $w\in X$ such that $\Phi(w)\leq r$, owing to (\ref{ineq}), we get
$$
\Psi(w)\leq\|K\|_{\infty}\Big((2r)^{1/2}S_1a_1+\frac{2^{q/2}S_q^qa_2}{q}r^{q/2}\Big).
$$
Therefore
\begin{eqnarray}\label{hh}
\sup_{w\in\Phi^{-1}(]-\infty,r])}\Psi(w)\leq \|K\|_{\infty}\Big((2r)^{1/2}S_1a_1+\frac{2^{q/2}S_q^qa_2}{q}r^{q/2}\Big).
\end{eqnarray}
Next, we set
\begin{equation}\label{EE7} \notag
r_{1}=\frac{\|\alpha\|_{L^1(\mathcal{M})}}{2}\gamma_{1}^{2}, \quad r_{2}=\frac{\|\alpha\|_{L^1(\mathcal{M})}}{2}\gamma_{2}^{2}, \quad \mbox{and} \quad w_{\delta}(\sigma)=\delta, \,\,\,\mbox{ for every }\; \sigma\in\mathcal{M}.
\end{equation}

\noindent Clearly $w_\delta\in X$ and we have
\begin{eqnarray}\label{norm}
\Phi(w_\delta)=\frac{1}{2}\Big(\int_{\mathcal{M}}|\nabla w_\delta(\sigma)|^2d\sigma_g+\int_{\mathcal{M}}\alpha(\sigma)w_\delta(\sigma)^2d\sigma_g\Big)=\frac{\delta^2}{2}\|\alpha\|_{L^1(\mathcal{M})}.
\end{eqnarray}
Taking into account that $\gamma_1<\delta<\gamma_2$, by a direct computation,
we deduce that
$r_1<\Phi(w_\delta)<r_2$.
\noindent Moreover,
\begin{eqnarray}\label{h728}
\Psi(w_\delta)=\int_{\mathcal{M}} K(\sigma)F(w_\delta(\sigma))d\sigma_g=F(\delta)\|K\|_{L^1(\mathcal{M})}.
\end{eqnarray}
>From (\ref{hh}) it follows that
\begin{equation} \label{Gio1}
\sup_{w\in\Phi^{-1}(]-\infty, r_2[)}\Psi(w) \leq \|K\|_{\infty}\left((2r_2)^{1/2}S_1a_1+\frac{2^{q/2}S_q^qa_2}{q}r_2^{q/2}\right).
\end{equation}
as well as
\begin{equation} \label{Gio2}
\sup_{u\in\Phi^{-1}(]-\infty, r_1])}\Psi(u) \leq \|K\|_{\infty}\left((2r_1)^{1/2}S_1a_1+\frac{2^{q/2}S_q^qa_2}{q}r_1^{q/2}\right).
\end{equation}
Then $r_1<\Phi(w_\delta)<r_2$ and
$$\beta(r_1,r_2):=\inf_{v \in \Phi^{-1}(]r_1,r_2[)} \frac{\displaystyle\sup_{w \in \Phi^{-1}(]r_1,r_2[)} \Psi(w)-\Psi(v)}{r_2-\Phi(v)}\leq \frac{\displaystyle\sup_{w \in \Phi^{-1}(]-\infty,r_2[)} \Psi(w)-\Psi(w_\delta)}{r_2-\Phi(w_\delta)},$$
and
$$\begin{array}{ll}
\displaystyle \rho_2(r_1,r_2)&\displaystyle :=\sup_{v \in \Phi^{-1}(]r_1,r_2[)} \frac{\Psi(v)-\displaystyle\sup_{w \in \Phi^{-1}(]-\infty,r_1])} \Psi(w)}{\Phi(v)-r_1}\\ \ \\
&\displaystyle\geq \frac{\Psi(w_\delta)-\displaystyle\sup_{w \in \Phi^{-1}(]-\infty,r_1])} \Psi(w)}{\Phi(w_\delta)-r_1}.\end{array}$$
\noindent Hence, by using the notation \eqref{g}, and relations from \eqref{norm} to \eqref{Gio2} it follows that
$$
\beta(r_1,r_2)\leq 2a_{\gamma_2}(\delta),\,\,\,\,\,\,{\rm
and}\,\,\,\,\,\,\rho_2(r_1,r_2)\geq 2a_{\gamma_1}(\delta).
$$
Finally, hypothesis \eqref{EE1} yields
$$\beta(r_1,r_2)<\rho_2(r_1,r_2).$$
\noindent Now, from \cite[Proposition 2.1]{Bonanno}, the functional $J_\lambda$ satisfies $^{[r_1]}({\rm PS})^{[r_2]}$-condition for all
$r_1$ and $r_2$ with $r_1<r_2<+\infty$. Therefore, owing to Theorem \ref{T2}, for each $\lambda
\in \left]\displaystyle
\frac{1}{2a_{\gamma_1}(\delta)},\frac{1}{2a_{\gamma_2}(\delta)}\right[$,
the functional $J_\lambda$ admits at least one critical point
$w_{0,\lambda}$ such that
$$r_1 < \Phi (w_{0,\lambda}) < r_2,$$ that is
$$\|\alpha\|_{L^1(\mathcal{M})}^{1/2}\gamma_{1}<\|w_{0,\lambda}\|_{{H_\alpha ^2}}<\|\alpha\|_{L^1(\mathcal{M})}^{1/2}\gamma_{2}.$$
The last part of our result is achieved from the above inequalities and taking into account relation \eqref{equivalance}.
\end{pf}

\indent We now point out the following consequence of Theorem \ref{TT1}.

\begin{corollary}\label{CC2}
Let $f:\erre\to\erre$ be a continuous function such that condition $({\rm h}_\infty)$ holds and assume that there exist two positive constants $\gamma$ and $\delta$, with $\gamma>\delta$, for which
\begin{equation} \label{F1}
    \frac{\displaystyle F(\delta)}{\delta^2}>\frac{A(\gamma)}{q\gamma^2\|K\|_{L^1(\mathcal{M})}}.
\end{equation}
Then, for each parameter $\lambda$ belonging to
$$
\left]\dfrac{\|\alpha\|_{L^1(\mathcal{M})}\delta^2}{\displaystyle 2\|K\|_{L^1(\mathcal{M})} F(\delta)},\dfrac{q\|\alpha\|_{L^1(\mathcal{M})}\gamma^2}{\displaystyle 2A(\gamma)} \right[,
$$
the problem $(P_{\lambda})$ admits at least one non-trivial weak solution $w_{0,\lambda}\in H_1^2({\mathcal{M}})$, such that $$\|w_{0,\lambda}\|_{{H_1 ^2}}<\displaystyle\frac{\|\alpha\|_{L^1(\mathcal{M})}^{1/2}\gamma}{\min\{1,{\displaystyle\essinf_{\sigma\in{{\mathcal{M}}}}\alpha(\sigma)^{1/2}}\}}.$$
\end{corollary}

\begin{pf}
Our aim is to apply Theorem \ref{TT1}. To this end we pick $\gamma_1=0$ and $\gamma_2:=\gamma$.
 Bearing in mind (\ref{g}), we obtain
 $$
 a_{\gamma}(\delta)=\frac{\displaystyle A(\gamma)-q\|K\|_{L^1(\mathcal{M})} F(\delta)}{\|\alpha\|_{L^1(\mathcal{M})}(\gamma^2-\delta^2)q},
 $$
 as well as
 $$
 a_0(\delta)=\frac{\displaystyle \|K\|_{L^1(\mathcal{M})}F(\delta)}{\delta^2\|\alpha\|_{L^1(\mathcal{M})}}.
 $$
 Now, inequality \eqref{F1}, immediately yields
 $$
 a_\gamma(\delta)<a_0(\delta).
 $$
 Hence, Theorem \ref{TT1} ensures the conclusion.
\end{pf}

Here is a direct result obtained by using Corollary \ref{CC2}.

\begin{theorem}\label{ultimo}
Let $f:\erre\to\erre$ be a continuous function such that condition $({\rm h}_\infty)$ holds
 and assume that
 \begin{equation}\tag{${\rm h}_{0,F}$}\label{EC3}
\lim_{\xi\rightarrow0^+} \frac{\displaystyle F(\xi)}{\xi^2}=+\infty.
\end{equation}
\noindent Furthermore, for each $\gamma>0$, set
$$
\lambda^\star_\gamma:=\frac{q\|\alpha\|_{L^1(\mathcal{M})}}{2}\dfrac{\gamma^2}{\displaystyle A(\gamma)}.
$$
Then, for every $\lambda\in ]0,\lambda^\star_\gamma[$,
the problem $(P_{\lambda})$ admits at least one non-trivial weak solution $w_{0,\lambda}\in H_1^2({\mathcal{M}})$. Moreover, we have
$$
\lim_{\lambda\rightarrow 0^+}\|w_{0,\lambda}\|_{{H_1^2}}=0,
$$
and the map
$$
\lambda\mapsto \frac{\|w_{0,\lambda}\|_{{H_\alpha^2}}^2}{2}-\lambda\int_{\mathcal{M}} K(\sigma)\left(\int_0^{w_{0,\lambda}(\sigma)}f(t)dt\right)d\sigma_g,
$$
is negative and strictly decreasing in $]0,\lambda^{\star}_\gamma[$.
\end{theorem}
\begin{pf}
Fix $\gamma>0$ and $\lambda\in ]0,\lambda^{\star}_\gamma[$. From
$({\rm h}_{0,F})$ there exists a positive constant $\delta$ with $\delta<\gamma$ such that
$$
\dfrac{\|\alpha\|_{L^1(\mathcal{M})}\delta^2}{\displaystyle 2\|K\|_{L^1(\mathcal{M})}F(\delta)}<\lambda<\dfrac{q\|\alpha\|_{L^1(\mathcal{M})}\gamma^2}{\displaystyle 2A(\gamma)}.
$$
Hence, owing to
Corollary \ref{CC2},
the problem $(P_{\lambda})$ admits at least one non-trivial weak solution $w_{0,\lambda}\in X$, such that $$\|w_{0,\lambda}\|_{{H_\alpha ^2}}<\|\alpha\|_{L^1(\mathcal{M})}^{1/2}\gamma.$$
Then, for every $\lambda\in ]0,\lambda^{\star}_{\gamma}[$, there exists at least one non-trivial weak solution $w_{0,\lambda}\in \Phi^{-1}(]0,r_2[)$ of the problem $(P_{\lambda})$ and
\begin{equation}\label{bounded0}
\|w_{0,\lambda}\|_{{H_1^2}}< \displaystyle\frac{\|\alpha\|_{L^1(\mathcal{M})}^{1/2}\gamma}{\min\{1,{\displaystyle\essinf_{\sigma\in{{\mathcal{M}}}}\alpha(\sigma)^{1/2}}\}},
\end{equation}
\noindent for every $\lambda\in ]0,\lambda^{\star}_{\gamma}[$.\par

\noindent Therefore, from $({\rm h}_{\infty})$, taking into account \eqref{ineq} and \eqref{bounded0}, it follows that there exists a positive constant $C^{\alpha, K}_{a_1,a_2,q}$ such that
\begin{equation}\label{boundedz}
\left|\int_{\mathcal{M}} K(\sigma)f(w_{0,\lambda}(\sigma))w_{0,\lambda}(\sigma)d\sigma_g\right|\leq C^{\alpha, K}_{a_1,a_2,q},
\end{equation}
\noindent for every $\lambda\in ]0,\lambda^{\star}_\gamma[$.\par
\noindent Now, $J'_\lambda(w_{0,\lambda})=0$, for every $\lambda\in ]0,\lambda^{\star}_\gamma[$ and in particular $J'_\lambda(w_{0,\lambda})(w_{0,\lambda})=0$, that is,
$$
\|w_{0,\lambda}\|_{{H_\alpha^2}}^2=\lambda\int_{\mathcal{M}} K(\sigma)f(w_{0,\lambda}(\sigma))w_{0,\lambda}(\sigma)d\sigma_g,
$$
\noindent for every $\lambda\in ]0,\lambda^{\star}_\gamma[$.\par
\noindent Then, from (\ref{boundedz}), it follows that
$$
\lim_{\lambda\rightarrow 0^+}\|w_{0,\lambda}\|_{{H_\alpha^2}}^2=\lim_{\lambda\rightarrow 0^+}\lambda\Psi'(w_{0,\lambda})(w_{0,\lambda})=0,
$$
\noindent that implies
$\displaystyle
\lim_{\lambda\rightarrow 0^+}\|w_{0,\lambda}\|_{{H_1^2}}=0.$\par
Finally, we claim that the mapping $\lambda\mapsto J_{\lambda}(w_{0,\lambda})$ is negative and strictly decreasing in $]0,\lambda^{\star}_{\gamma}[$. Indeed,  the restriction
of the functional $J_{\lambda}$ to
$\Phi^{-1}(]0,r_2[)$, where $r_{2}:=({\|\alpha\|_{L^1(\mathcal{M})}}/{2})\gamma_{2}^{2}$, admits a global minimum, which is a
critical point $($local minimum$)$ of $J_{\lambda}$ in $X$. Moreover, since $w_\delta:=\delta\in\Phi^{-1}(]0,r_2[)$ and
$$
\frac{\Phi(w_\delta)}{\Psi(w_\delta)}=\dfrac{\|\alpha\|_{L^1(\mathcal{M})}\delta^2}{\displaystyle 2\|K\|_{L^1(\mathcal{M})}F(\delta)}<\lambda,
$$
we have
$$
J_\lambda(w_{0,\lambda})\leq J_\lambda(w_{\delta})=\Phi(w_\delta)-\lambda\Psi(w_\delta)<0.
$$
 Next, we observe that
$$
J_{\lambda}(w)=\lambda\left(\frac{\Phi(w)}{\lambda}-\Psi(w)\right),
$$
for every $u\in X$ and fix $0<\lambda_1<\lambda_2<\lambda^{\star}_{\gamma}$. Set
$$
m_{\lambda_1}:=\left(\frac{\Phi(w_{0,\lambda_1})}{\lambda_1}-\Psi(w_{0,\lambda_1})\right)=\inf_{w\in \Phi^{-1}(]0,r_2[)}\left(\frac{\Phi(w)}{\lambda_1}-\Psi(w)\right),
$$
and
$$
m_{\lambda_2}:=\left(\frac{\Phi(w_{0,\lambda_2})}{\lambda_2}-\Psi(w_{0,\lambda_2})\right)=\inf_{w\in \Phi^{-1}(]0,r_2[)}\left(\frac{\Phi(w)}{\lambda_2}-\Psi(w)\right).
$$
Clearly, as claimed before, $m_{\lambda_i}<0$ (for $i=1,2$), and $m_{\lambda_2}\leq m_{\lambda_1}$ thanks to $\lambda_1<\lambda_2$.
Then the mapping $\lambda\mapsto J_{\lambda}(w_{0,\lambda})$ is strictly decreasing in $]0,\lambda^{\star}_{\gamma}[$ owing to
$$
J_{\lambda_2}(w_{0,\lambda_2})=\lambda_2m_{\lambda_2}\leq \lambda_2m_{\lambda_1}<\lambda_1m_{\lambda_1}=J_{\lambda_1}(w_{0,\lambda_1}).
$$ This concludes the proof.
\end{pf}

\begin{remark}\label{moregeneral}
\rm{A careful analysis of the proof of Theorem \ref{ultimo} ensures that the result still remains true after replacing condition $({\rm h}_{0,F})$
with the more general assumption at zero
\begin{equation}\tag{${\rm h}_{0,F}''$}\label{G4}
\limsup_{\xi\rightarrow0^+} \frac{\displaystyle F(\xi)}{\xi^2}=+\infty.
\end{equation}
Moreover, if $f$ has the following asymptotic behaviour
\begin{equation}\tag{${\rm h}_{0}$}\label{G5}
\lim_{t\rightarrow0^+} \frac{\displaystyle f(t)}{t}=+\infty,
\end{equation}
then, obviously, hypothesis $({\rm h}_{0,F})$ is trivially verified. Hence, it is easy to see that Theorem \ref{introSfera} in Introduction immediately follows from the above remark and Theorem \ref{ultimo}.
}
\end{remark}

\begin{remark}\label{maximalinterval}
\rm{
In other words, Theorem \ref{ultimo} ensures that if $f$ has the global growth given by $({\rm h}_\infty)$ and the asymptotic condition at zero $({\rm h}_{0,F})$ is verified then, for every parameter $\lambda$ belonging to the real interval $\Lambda_{\mathcal{M}}:=]0,\lambda^{\star}[$, where
$$
\lambda^{\star}:=\frac{q\|\alpha\|_{L^1(\mathcal{M})}}{2}\sup_{\gamma>0}\dfrac{\gamma^2}{\displaystyle A(\gamma)},
$$
\noindent problem $(P_{\lambda})$ admits at least one non-trivial solution $w_{0,\lambda}\in H_1^2({\mathcal{M}})$. Moreover $
{\|w_{0,\lambda}\|_{{H_1^2}}}\rightarrow 0,$ as $\lambda\rightarrow 0^+$.
 Moreover, a straightforward computation shows that
 $$
 \lambda^{\star}:=
 \left\{
\begin{array}{l}
+\infty \quad\, {\rm if} \quad 1<q<2 \\
\displaystyle\frac{1}{\|K\|_{\infty}S^2_2a_2} \quad {\rm if} \quad q=2 \\
\displaystyle\frac{q\|\alpha\|_{L^1(\mathcal{M})}^{1/2}\overline{\gamma}_{\max}}{2\|K\|_{\infty}(qS_1a_1+\|\alpha\|_{L^1(\mathcal{M})}^{(q-1)/2}{S_q^qa_2}\overline{\gamma}_{\max}^{q-1})} \quad {\rm if} \quad q\in \left]2,\frac{2d}{d-2}\right[, \\
\end{array}
\right.
 $$
\noindent where
$$
\overline{\gamma}_{\max}:=\frac{1}{\|\alpha\|_{L^1(\mathcal{M})}^{1/2}}\left(\frac{qS_1a_1}{(q-2)S_q^qa_2}\right)^{1/(q-1)}.
$$
We also note that in the case $q\in \left]2,2^*\right[$, we have $$\|u_{0,\lambda}\|_{{H_1^2}}<\left(\frac{qS_1a_1}{(q-2)S_q^qa_2}\right)^{1/(q-1)}\left(\displaystyle\frac{1}{\min\{1,{\displaystyle\essinf_{\sigma\in{{\mathcal{M}}}}\alpha(\sigma)^{1/2}}\}}\right),$$ uniformly for every $\lambda \in \Lambda_{\mathcal{M}}$.
}
\end{remark}
\begin{remark} \label{direct}
{\rm From the above observation, it follows that if $f$ is a sublinear function at infinity, Theorem \ref{ultimo} ensures that,
for each $\lambda>0$, the problem $(P_{\lambda})$ admits at least one non-zero weak solution. We explicitly observe that, in this case, also the classical direct methods theorem ensures the existence of at least one weak solution. However, in this case, it may be zero.}
\end{remark}
\begin{remark}\label{costanti}
\rm{In our context,
a concrete upper bound for the constants $S_q$ in Theorem \ref{ultimo} is essential for a concrete evaluation of the interval $\Lambda_{\mathcal{M}}$.
  In the case $({\mathcal{M}},g)=({\mathbb{S}}^{d},h)$, if $q\in[1,2d/(d-2)[$, we have
\begin{equation}\label{inequality2}
\displaystyle S_q\leq S^{\star}_q:=\frac{\kappa_q}{\min\left\{1,\displaystyle\essinf_{\sigma\in {\mathbb{S}}^{d}}\alpha(\sigma)^{1/2}\right\}},
\end{equation}
\noindent where, we set
\begin{equation*}
\displaystyle \kappa_q:=\left\{
\begin{array}{l}\omega_d^{\frac{2-q}{2q}}\quad\, {\rm{if}}\quad
q\in [1,2[, \\\\
{\displaystyle\max\left\{
\displaystyle\left(\frac{q-2}{d\omega_d^{\frac{q-2}{q}}}
\right)^{1/2},
\frac{1}{\omega_d^{\frac{q-2}{2q}}}
\right\}\quad {\rm{if}}\quad q\in \left[2,\frac{2d}{d-2}\right[.}\\
\end{array}
\right.
\end{equation*}
Indeed, in Beckner \cite{Be}, it is proved that for every $2\leq q< 2d/(d-2)$ and any $w\in H^{2}_1({\mathbb{S}}^{d})$,
$$
\left(\int_{{\mathbb{S}}^{d}}|w(\sigma)|^qd\sigma_h\right)^{2/q}\leq \frac{q-2}{d\omega_d^{1-2/q}}\int_{{\mathbb{S}}^{d}}|\nabla w(\sigma)|^2d\sigma_h+\frac{1}{\omega_d^{1-2/q}}\int_{{\mathbb{S}}^{d}}w(\sigma)^2d\sigma_h,
$$
\noindent see also, for instance, Theorem 4.28 in Hebey \cite{Hebey}. Hence,
$$
\|w\|_{L^q({\mathbb{S}}^{d})}\leq \displaystyle\max\left\{
\displaystyle\left(\frac{q-2}{d\omega_d^{\frac{q-2}{q}}}
\right)^{1/2},
\frac{1}{\omega_d^{\frac{q-2}{2q}}}
\right\}\left(\int_{{\mathbb{S}}^{d}}|\nabla w(\sigma)|^2d\sigma_h+\int_{{\mathbb{S}}^{d}}w(\sigma)^2d\sigma_h\right)^{1/2},
$$
for every $w\in H^{2}_1({\mathbb{S}}^{d})$. Owing to (\ref{equivalance}) the desiderated statement follows.\par
\noindent On the other hand, if $q\in [1,2[$, as simple consequence of H\"{o}lder's inequality, it follows that
$$
\|w\|_{L^q({\mathbb{S}}^{d})}\leq \omega_d^{\frac{2-q}{2q}}\|w\|_{L^2({\mathbb{S}}^{d})},\qquad\mbox{for all}\ w\in L^2({\mathbb{S}}^{d}).
$$
The thesis is achieved taking into account that
$$
\|w\|_{L^2({\mathbb{S}}^{d})}\leq \|w\|_{H^2_1}\leq \frac{\|w\|_{H^2_\alpha}}{\min\left\{1,\displaystyle\min_{\sigma\in {\mathbb{S}}^{d}}\alpha(\sigma)^{1/2}\right\}},
$$
for every $w\in H^{2}_1({\mathbb{S}}^{d})$.
Note also that if $d\geq 4$, it follows that
$$
q-2<\frac{2d}{d-2}-2\leq d.
$$
In this case, clearly
$$
\frac{q-2}{\omega_d^{\frac{q-2}{q}}}
<
\frac{d}{\omega_d^{\frac{q-2}{q}}},
$$
\noindent and we have $\kappa_q=\omega_d^{\frac{2-q}{2q}},$ for every $q\in [1,2^{*}[$. Consequently, if $d\geq 4$, we obtain
$$
S^{\star}_q:=\frac{\omega_d^{\frac{2-q}{2q}}}{\min\left\{1,\displaystyle\essinf_{\sigma\in {\mathbb{S}}^{d}}\alpha(\sigma)^{1/2}\right\}},
$$
for every $q\in [1,2^{*}[$.
}
\end{remark}

\begin{remark} \label{nonnegativity}
{\rm
  We observe that if $f$ is a non-negative function our results guarantee that the attained weak solution is non-negative. From our goal, let $w_0$ be a weak solution of problem $(P_{\lambda})$. Arguing by contradiction, assume that the set ${\mathcal{M}}_0:=\big\{\sigma \in {\mathcal{M}} : w_0(\sigma)<0\big\}$ has positive Riemannian measure. Put $\overline w(\sigma):=\min\{0, w_0(\sigma)\}$ for all $\sigma \in {\mathcal{M}}$. Clearly, $\overline w \in H_1 ^2({\mathcal{M}})$ and
$$
\int_{{\mathcal{M}}} \langle
\nabla w(\sigma),\nabla \overline{w}(\sigma) \rangle_g
d\sigma_g+\int_{{\mathcal{M}}} \alpha(\sigma)\langle w(\sigma),\overline{w}(\sigma)\rangle_g d\sigma_g
$$
$$
-\lambda\int_{\mathcal{M}} K(\sigma)f(w(\sigma))\overline{w}(\sigma)d\sigma_g=0,\quad
$$
that is,
$$
\int_{{\mathcal{M}}_0} |\nabla w_0(\sigma)|^2 d\sigma_g+\int_{{\mathcal{M}}_0}\alpha(\sigma)w_0(\sigma)^2 d\sigma_g
=\lambda \int_{{\mathcal{M}}_0} K(\sigma)f(w(\sigma)){w}_0(\sigma)d\sigma_g\leq 0.
$$
Hence
$$\int_{{\mathcal{M}}_0} |\nabla w_0(\sigma)|^2 d\sigma_g+\int_{{\mathcal{M}}_0}\alpha(\sigma)w_0(\sigma)^2 d\sigma_g=0.$$
Then, $w_0=0$ almost everywhere in ${\mathcal{M}}_0$. This is not possible by the definition of ${\mathcal{M}}_0$, so it follows that $w_0$ is non-negative. It is also evident that the above assertion remains valid requiring only that $f$ is non-negative in $[0,+\infty[$.
}
\end{remark}
The following example deals with a nonlinear problem on the unit sphere endowed with the natural metric $h$ and involving a nonlinearity with subcritical growth.
\begin{example} \label{Ex1}
{\rm  Let $({\mathbb{S}}^{d},h)$ with $d\geq 3$ and $K\in \Lambda_+({{\mathbb{S}}^{d}})$. Further, consider the following equation
$$-\Delta_{h} w+ w= \lambda K(\sigma)(|w|^{r-2}w+|w|^{s-2}w), \eqno{(\widetilde{P}_{\lambda})}$$
\noindent for every $\sigma\in {{{\mathbb{S}}^{d}}}$ and $\ w\in H_1^2({{\mathbb{S}}^{d}})$, where $1<r<2$ and $2<s< 2^*$. Then, for every
$$\lambda\in\left]0,\frac{s\omega_d^{1/2}\overline{\gamma}_{\max}}
{4\|K\|_\infty(sS_1+\omega_d^{(s-1)/2}{S_s^s}\overline{\gamma}_{\max}^{s-1})}\right[,$$
\noindent where
$$
\overline{\gamma}_{\max}:=\frac{1}{\omega_d^{1/2}}\left(\frac{S_1}{S_s^s}\left(\frac{s}{s-2}\right)\right)^{1/(s-1)},
$$
\noindent the problem $(\widetilde{P}_{\lambda})$ admits at least one non-negative (and non-trivial)
weak solution $w_{0,\lambda}\in H_1^2({{\mathbb{S}}^{d}})$ such that
$$
\|w_{0,\lambda}\|_{{H_1^2}}<\left(\frac{S_1}{S_s^s}\left(\frac{s}{s-2}\right)\right)^{1/(s-1)},
$$
and $ \lim_{\lambda\rightarrow 0^+}\|w_{0,\lambda}\|_{{H_1^2}}=0. $\par
\noindent To prove this, we can apply Theorem \ref{ultimo} with
$$
f(t):=|t|^{r-2}t+|t|^{s-2}t,
$$
\noindent for every $t\in\erre$. Indeed, it is easy to
verify that
$$
|f(t)|\leq
2(1+|t|^{s-1}),\,\,\,\,\forall\,
t\in\erre.
$$
Moreover, a direct computation shows that
$$
\lim_{\xi\rightarrow0^+} \frac{\displaystyle
F(\xi)}{\xi^2}\geq
\frac{1}{r}\left(\lim_{\xi\rightarrow0^+}
\frac{1}{\xi^{2-r}}\right)=+\infty.
$$
\noindent Hence, all the assumptions of Theorem \ref{ultimo} are
verified and the conclusion follows. }
\end{example}

\begin{remark} \label{remcoercivity}
{\rm
We point out that the energy functional $J_\lambda$ associated to problem $(\widetilde{P}_{\lambda})$ is unbounded from below. In fact, if we fix $v\in H_1^2({{\mathbb{S}}^{d}})$ and $\tau\in\erre$, then
$$
J_\lambda(\tau v)\leq\frac{\tau^2}{2}\|v\|^2_{H_\alpha^2}-\lambda\left[\frac{\tau^r}{r}\|v\|^{r}_{L^r({{\mathbb{S}}^{d}})}+\frac{\tau^s}{s}\|v\|^{s}_{L^s({{\mathbb{S}}^{d}})}\right]\essinf_{\sigma\in{{\mathbb{S}}^{d}}}K(\sigma).
$$
\noindent So, as $r<2<s$, it follows that
$$
\lim_{\tau\rightarrow+\infty}J_\lambda(\tau v)=-\infty.
$$
Hence, as consequence, the functional $J_\lambda$ is not coercive. Hence, the classical Tonelli's method cannot be applied to the above case.
}
\end{remark}


\begin{example}
\rm{Let $\alpha,K\in\Lambda_+({\mathbb{S}}^d)$. Owing to Theorem \ref{ultimo} and taking into account Remark \ref{maximalinterval}, the following elliptic parametric problem
$$-\Delta_{h}w+\alpha(\sigma)w= \lambda
K(\sigma)\sqrt{|w|},\qquad \sigma\in {\mathbb{S}}^d,\, w\in H_1^2({\mathbb{S}}^d),\eqno(S_\lambda^{\alpha})
$$
 admits at least one non-trivial weak solution for all $\lambda>0$. In this case Theorem 1.1 obtained in \cite{K0} cannot be applied just because the real function defined by $f(t)=\sqrt{|t|}$, for every $t\in\erre$, does not belong to $\mathcal{F}$.}
\end{example}

\begin{remark}
\rm{We also point out that contributions on the existence of multiple solutions for elliptic problems on the spherical case are contained in Krist\'aly \cite{K1}; see also the related paper Krist\'{a}ly and Marzantowicz \cite{KM}.}
\end{remark}

In the sequel we prove how the previous results can be used in order to pass from the existence of at least one nontrivial solution to the existence of at least two nontrivial solutions. This goal will be achieved making use of the particular nature of the first solution, namely the local minimum of the associated energy functional. This information will be used to assure the existence of a second solution as a critical point of mountain pass type. In this direction, we begin with the following theorem, where the celebrated Ambrosetti-Rabinowitz condition is required. As usual, this assumption plays a crucial role in proving that every Palais-Smale sequence is bounded, as well as that the so called `mountain pass geometry' is satisfied.

\begin{theorem}\label{CAR}
Let $f:\erre\to\erre$ be a continuous function such that
\begin{equation}\tag{${\rm h}_{\infty}'$}\label{Su}
|f(t)|\leq a_1+a_2|t|^{q-1},\,\,\,\,\,\forall\, t\in \erre,
\end{equation}
for some non-negative constants $a_1,a_2$, where $q\in ]2, 2^*[$. Furthermore, assume that condition $({\rm h}_{0,F}'')$ holds in addition to:
\begin{itemize}
  \item [(AR)]
  there are constants $\mu>2$ and $r>0$ such that, for all $|\xi|\geq
  r$,
    $$
    0<\mu F(\xi)\leq \xi f(\xi).
    $$
\end{itemize}
Then, for each $\lambda\in\Lambda_{\mathcal{M}}$, the problem $(P_{\lambda})$
admits at least two weak solutions.
\end{theorem}
\begin{pf}
Fix $\lambda\in \Lambda_{\mathcal{M}}$. Owing to $({\rm h}_{\infty}')$ and $({\rm h}_{0,F}'')$, Theorem
\ref{ultimo} ensures that the problem
$(P_{\lambda})$ admits at least one weak non-trivial solution $w_1$
which is a local minimum of the functional $J_\lambda$ as defined in
the proof of Theorem \ref{TT1}. In view of assumption $(\rm AR)$, reasoning in a standard way, it is possible to verify that every Palais-Smale sequence is bounded. This, leads to the fact that $J_\lambda$ satisfies the classical Palais-Smale condition by the fact that $\Phi'$ is a linear isomorphism, while $\Psi'$ is compact; see \cite[Proposition 3.8]{G}. Moreover, it is well known that, again by $(\rm AR)$, there exist suitable (positive) constants $\gamma_1,\gamma_2$ with
\begin{equation}\label{superquadratic}
F(\xi)\geq \gamma_1 |\xi|^\mu-\gamma_2,\quad  \forall\; \xi\in\erre.
\end{equation}
Let $\{\xi_n\}$ be a sequence in $\erre$ such that $\xi_n\to +\infty$ and consider the related sequence of functions $w_n(\sigma):=\xi_n$ for every $\sigma\in \mathcal{M}$ which belongs to $X$. Obviously, in view of (\ref{superquadratic}), we have
$$
J_\lambda(w_n)\leq\frac{\xi_n^2}{2}\|\alpha\|_{L^1({\mathcal{M}})}-\lambda\left(\gamma_1\xi_n^{\mu}-\gamma_2\right)\|K\|_{L^1(\mathcal{M})},
$$
\noindent that is
$$
\liminf_{\|w\|_{X}\to \infty}J_\lambda(w)=-\infty,
$$
and there exists some $w_0\in X$ such that
\begin{equation}\label{u0<u1}
J_\lambda(w_0)<J_\lambda(w_1).
\end{equation}

\noindent Now, we can assume that $w_1$ is a strict
local minimum for $J_\lambda$ in $X$,
otherwise there exist infinitely many nontrivial critical points of $J_\lambda$. Hence, we can apply the classical mountain pass theorem and obtain a second critical point $w_2\in X$ such that $J_\lambda(w_2)>J_\lambda(w_1)$, that is $w_1\neq w_2$ and the proof is complete.
\end{pf}

\begin{remark}\label{comp}
\rm{The existence of two solutions for the problem $(P_\lambda)$ has been investigated by Krist\'aly, R\u adulescu and Varga in \cite[Theorem 9.4, p. 222]{KRV} and by Krist\'aly, R\u adulescu in \cite[Theorem 1.1]{KR}. Moreover, very recently, Krist\'aly in \cite{K0} studied bifurcation effects for a sublinear problem $(P_{\lambda,\mu})$ defined on a compact Riemannian manifold $\mathcal{M}$ without boundary. All the above cited results cannot be applied to our cases due to the asymptotic condition $(\rm h_0)$ that we assume at zero. Finally, we observe that assuming $f(0)\neq 0$, Corollary \ref{CAR} ensures the existence of at least two non-zero solutions for problem $(P_\lambda)$; see also Corollary \ref{cork} in the sequel.}
\end{remark}

As usual, condition $(\rm AR)$ plays a crucial role in proving that every Palais-Smale sequence is bounded, as well as that the so called `mountain pass geometry' is satisfied. However, even dealing with different problems than ours, several authors studied more general or different assumptions that still allow to apply min-max methods in order to assure the existence of critical points. Here we show a result that moves in this direction. We recall that a $C^1$ functional $J:X\rightarrow \erre$ defined on a real Banach space $X$ satisfies the Cerami condition (briefly (C)) if\par
\begin{itemize}
\item[(C)]\textit{Every sequence $\{x_n\}$ in $X$ such that $\{J_\lambda(x_n)\}$ is bounded and
$$
(1+\|x_n\|_X)\|J_\lambda'(x_n)\|_{X^{*}}\to 0,$$
admits a  strongly convergent subsequence in $X$. A sequence $\{x_n\}$ that satisfies the above conditions is called a \textit{Cerami sequence}.}
\end{itemize}
We recall here, for reader's convenience, the following Lemma due to Liu; see \cite[Lemma 2.5]{Liu}.
\begin{lemma}\label{lemmacerami}
Let $f:\erre\rightarrow\erre$ be a continuous function and set
$${\widetilde{F}}(\xi):=\xi f(\xi)-2F(\xi), \eqno{(\mathcal{F})}$$
\noindent for every $\xi\in\erre$.
Assume that
\begin{itemize}
\item[$(f_1)$] there exists $\rho>0$ such that the function
$$
g(t):=\frac{f(t)}{t},
$$
is non-decreasing in $t\geq \rho$ and non-increasing in $t\leq -\rho$.
\end{itemize}
\noindent Then, the real mapping ${\widetilde{F}}$, defined in $(\mathcal{F})$, is non-decreasing in $t\geq \rho$ and non-increasing in $t\leq -\rho$. In particular, there is a constant $C_1>0$ such that
\begin{equation}\label{20}
{\widetilde{F}}(\xi_1)\leq {\widetilde{F}}(\xi_2)+C_1,
\end{equation}
for $0\leq \xi_1\leq \xi_2$ or $\xi_2\leq \xi_1\leq 0$.
\end{lemma}
\begin{pf}
Let us consider only the case $\xi_1\leq \xi_2\leq t$. We have
\begin{eqnarray*}\notag
  {\widetilde{F}}(\xi_2)-{\widetilde{F}}(\xi_1)   &=& 2\left[\frac{(f(\xi_2)t-f(\xi_1)\xi_1)-(F(\xi_2)-F(\xi_1))}{2}\right]\nonumber\\
                    &=& 2\Big [\int_\rho^{\xi_2}g(\xi_2)\tau d\tau-\int_\rho^{\xi_1} g(\xi_1)\tau d\tau-\int_{\xi_1}^{\xi_2} g(\tau)\tau d\tau \nonumber\\
                    &-& \frac{g(\xi_2)\rho^2}{2}-\frac{g(\xi_1)\rho^2}{2}\Big]\nonumber\\
                    &=& 2\Big[\int_{\xi_1}^{\xi_2}(g(\xi_2)-g(\tau))\tau d\tau+\int_\rho^{\xi_1}(g(\xi_2)-g(\xi_1))\tau d\tau\nonumber\\
                    &+& \frac{\rho^2}{2}(g(\xi_2)-g(\xi_1))\Big]\geq 0.\nonumber
\end{eqnarray*}
The case $\xi_2\leq \xi_1\leq -\rho$ is similar. Furthermore, condition implies that $\widetilde{F}\in C^0(\erre;\erre)$ and
$$
C_1:=1+\max_{\xi\in [-\rho,\rho]}{\widetilde{F}}(\xi)-\min_{\xi\in [-\rho,\rho]}{\widetilde{F}}(\xi)<+\infty.
$$
With this positive constant $C_1$ it is easy to see that condition \eqref{20} holds.
\end{pf}

As consequence of Theorem \ref{ultimo} and the above Lemma we obtain the following multiplicity result.
\begin{theorem}\label{TheoremCerami}
Let $f:\erre\to\erre$ be a continuous function such that conditions $({\rm h}_{\infty}')$ and $(f_1)$ hold in addition to
\begin{equation}\tag{${\rm h}_{\infty, F}$}\label{Gigio}
\lim_{|\xi|\rightarrow \infty} \frac{\displaystyle F(\xi)}{\xi^2}=+\infty.
\end{equation}
 Furthermore, assume that
\begin{equation}\tag{${\rm h}_{0}'$}\label{Gigio36}
\lim_{\xi\rightarrow 0^+} \frac{\displaystyle F(\xi)}{\xi^2}=+\infty.
\end{equation}
Then, for each $\lambda\in\Lambda_{\mathcal{M}}$, the problem $(P_{\lambda})$
admits at least two weak solutions. If, in addition, $f(0)\neq 0$ the attained solutions are non-zero.
\end{theorem}

\begin{pf}
Let $X:=H_1^2({\mathcal{M}})$. Fix $\lambda\in\Lambda_{\mathcal{M}}$ and argue as in the proof of Theorem \ref{CAR} in order to assure the existence of a first non-trivial weak solution. Moreover, in view of assumption $({\rm h}_{\infty, F})$, it is possible to verify that
\begin{equation}\label{liminf=-infty}
\liminf_{\|w\|_X\to +\infty}J_\lambda(w)=-\infty.
\end{equation}
Indeed, for
$$\eta>\displaystyle\frac{\|\alpha\|_{L^1({\mathcal{M}})}}{2\lambda\|K\|_{L^1({\mathcal{M}})}},$$ there exists $\tau>0$ such that $F(\xi)>\eta\xi^2$ for every $|\xi|>\tau$. Hence, if $\{\xi_n\}$ is a sequence in $\erre$ with $\xi_n\to +\infty$ and we consider the sequence in $X$ defined by putting $w_n(\sigma)=\xi_n$ for every $\sigma\in \mathcal{M}$ we have
$$
J_\lambda(w_n)=\frac{\|\alpha\|_{L^1({\mathcal{M}})}}{2} \xi_n^2 -\lambda \|K\|_{L^1({\mathcal{M}})} F(\xi_n)< \left(\frac{\|\alpha\|_{L^1({\mathcal{M}})}}{2}-\lambda \|K\|_{L^1({\mathcal{M}})} \eta \right) \xi_n^2,
$$
for every $n\in\enne$ large enough and (\ref{liminf=-infty}) follows from the choice of $\eta$. Hence, there exists $u_1\in X$ such that (\ref{u0<u1}) is true. To complete the proof, it is sufficient to prove that, from Lemma \ref{lemmacerami}, the functional $J_\lambda$ satisfies the (C)-condition. For our goal we argue following \cite[Lemma 2.5]{Liu}. Hence, let $\{w_n\}$ be a Cerami sequence of $J_\lambda$. We observe that it suffices to show that $\{w_n\}$ is bounded. Indeed, if $\{w_n\}$ is a bounded sequence in $X$  such that $\{J_\lambda(w_n)\}$ is bounded and $\|J'_\lambda(w_n)\|_{X^{*}}\rightarrow 0$, since $\Phi'$ is a linear isomorphism and $\Psi'$ is compact, passing to a subsequence if necessary, $\{w_n\}$ is strongly convergent in $X$; see \cite[Proposition 3.8]{G}. At this point, in order to prove the boundedness of $\{w_n\}$, we argue by contradiction. If $\{w_n\}$ is unbounded, up to a subsequence we may assume that there is $c\in \erre$ such that:
\begin{itemize}
\item[] $J_\lambda(w_n)\rightarrow c$
\item[] $\|w_n\|_{X}\rightarrow +\infty$
\item[] $\|w_n\|_{X}\|J'_\lambda(w_n)\|_{X^{*}}\rightarrow 0.$
\end{itemize}
In particular, since $J'_\lambda(w_n)(w_n)\rightarrow 0$ (as $n\rightarrow \infty$), we have
\begin{equation}\label{felix}
\lim_{n\rightarrow \infty}\int_{\mathcal{M}}K(\sigma)\frac{{\widetilde{F}}(w_n(\sigma))}{2}d\sigma_g=\frac{\displaystyle\lim_{n\rightarrow \infty} \left\{J_\lambda(w_n)-\frac{1}{2}J'_\lambda(w_n)(w_n)\right\}}{\lambda}=\frac{c}{\lambda}.
\end{equation}
Let $x_n:={w_n}/{\|w_n\|_{H_\alpha^2}}$, for every $n\in\enne$. Up to a subsequence, we have that
\smallskip
\begin{itemize}
\item[] $x_n\rightharpoonup x$ in $X$
\item[] $x_n \rightarrow x$ in $L^q(\mathcal{M})$
\item[] $x_n(\sigma)\rightarrow x(\sigma)$ a.e. $\sigma\in\mathcal{M}$.
\end{itemize}

\indent If $x=0$, we choose a sequence $\{t_n\}\subset [0,1]$ such that
$$
J_\lambda(t_nw_n)=\max_{t\in [0,1]} J_\lambda(tw_n).
$$
\noindent For any $m>0$, let $v_n:=2\sqrt{m}x_n$, for every $n\in\enne$.\par
\noindent At this point, owing to $v_n\rightarrow 0$ in $L^q(\mathcal{M})$ (so $v_n\rightarrow 0$ in $L^1(\mathcal{M})$) and
$$
\int_{\mathcal{M}}|K(\sigma)F(v_n(\sigma))|d\sigma_g\leq \|K\|_{\infty}\left(a_1\|v_n\|_{L^1(\mathcal{M})}+\frac{a_2}{q}\|v_n\|^{q}_{L^q(\mathcal{M})}\right)\rightarrow 0,
$$
as $n\rightarrow \infty$. Thus
$$
\lim_{n\rightarrow \infty}\left|\int_{\mathcal{M}}K(\sigma)F(v_n(\sigma))d\sigma_g\right|=0.
$$
So, for $n$ sufficiently large, ${2\sqrt{m}}/{\|w_n\|_{H_\alpha^2}}\in ]0,1[$, and
$$
\|v_n\|_{H_\alpha^2}^2=\|2\sqrt{m}x_n\|_{H_\alpha^2}^2=\left\|2\sqrt{m}\frac{w_n}{\|w_n\|_{H_\alpha^2}}\right\|_{H_\alpha^2}^2=4m.
$$
Hence, there exists $\nu\in\enne$ such that for evert $n\geq \nu$, we can write
\begin{equation*}
J_\lambda(t_nw_n):=\max_{t\in [0,1]} J_\lambda(tw_n)\geq J_\lambda(v_n)\geq 2m-\left|\int_{\mathcal{M}}K(\sigma)f(v_n(\sigma))d\sigma_g\right|\geq m.
\end{equation*}
\noindent Then, we have
\begin{equation}\label{limite}
J_\lambda(t_nw_n)\rightarrow +\infty,
\end{equation}
as $n\rightarrow \infty$. Now, since $J_\lambda(0)=0$ and $J_\lambda(w_n)\rightarrow c$, we deduce that $t_n\in ]0,1[$ and

\begin{eqnarray*}
   \|t_nw_n\|_{H_\alpha^2}^2 -\lambda\int_{\mathcal{M}} K(\sigma)f(t_nw_n(\sigma))t_nw_n(\sigma)d\sigma_g &=&  J'_\lambda(t_nw_n)(t_nw_n)\\
                    &=& t_n\frac{d}{dt}|_{t=t_n}J_\lambda(t_nw_n)=0.\nonumber
\end{eqnarray*}
Therefore, using relation \eqref{20} in Lemma \ref{lemmacerami} and the above computation, we deduce that
\begin{eqnarray*}\notag
 \int_{\mathcal{M}} K(\sigma)\frac{\widetilde{F}(w_n(\sigma))}{2}d\sigma_g   &\geq & \int_{\mathcal{M}} K(\sigma)\frac{\widetilde{F}(t_nw_n(\sigma))}{2}d\sigma_g -\frac{C_1}{2}\|K\|_{L^1(\mathcal{M})} \nonumber\\
                    &=& \frac{1}{2\lambda}\|t_nw_n\|^2_{H_\alpha^2}-\int_{\mathcal{M}}K(\sigma)F(t_nw_n(\sigma))d\sigma_g \nonumber\\
                    &-& \frac{C_1}{2}\|K\|_{L^1(\mathcal{M})}\nonumber\\
                    &=& \frac{J_\lambda(t_nw_n)}{\lambda}-\frac{C_1}{2}\|K\|_{L^1(\mathcal{M})}.\nonumber
\end{eqnarray*}
Therefore, since \eqref{limite} holds, it follows that
$$
\int_{\mathcal{M}} K(\sigma)\frac{\widetilde{F}(w_n(\sigma))}{2}d\sigma_g \rightarrow +\infty.
$$
This relation contradicts with \eqref{felix}.\par
If $x\neq 0$ the sub-manifold
$$
{\mathcal{M}}_{\Theta}:=\{\sigma\in {\mathcal{M}}:x(\sigma)\neq 0\},
$$
has positive Riemann measure. Since $|w_n(\sigma)|=|x_n(\sigma)|\|w_n\|_{H_\alpha^2}$ for every $\sigma\in \mathcal{M}$, for $\sigma\in {\mathcal{M}}_{\Theta}$, we have $|w_n(\sigma)|\rightarrow \infty$ and thanks to $({\rm h}_{\infty, F})$, we obtain
$$
\frac{F(w_n(\sigma))}{w_n(\sigma)^2}x_n(\sigma)^2\rightarrow +\infty.
$$
Hence, as $n\rightarrow \infty$, the Fatou's Lemma, implies
\begin{equation}\label{felix2}
\int_{\mathcal{M}_{\Theta}}\frac{F(w_n(\sigma))}{w_n(\sigma)^2}x_n(\sigma)^2d\sigma_g\rightarrow +\infty.
\end{equation}
Since $J_\lambda(w_n)\rightarrow c$, clearly
$$
\frac{1}{2}\|w_n\|^2_{H_\alpha^2}-\lambda\int_{\mathcal{M}}K(\sigma)F(w_n(\sigma))d\sigma_g-c=o(1),
$$
where $o(1)\rightarrow 0$ as $n\rightarrow \infty$.
Then
$$
\frac{1}{2}-\frac{c+o(1)}{\|w_n\|^2_{H_\alpha^2}}=\lambda\int_{\mathcal{M}}\frac{K(\sigma)F(w_n(\sigma))}{\|w_n\|_{H_\alpha^2}^2}d\sigma_g.
$$
Consequently, we can write
 \begin{eqnarray}\label{deco}
\frac{1}{2}-\frac{c+o(1)}{\|w_n\|^2_{H_\alpha^2}} &= & \lambda\int_{\mathcal{M}}\frac{K(\sigma)F(w_n(\sigma))}{\|w_n\|_{H_\alpha^2}^2}d\sigma_g \nonumber\\
                    &=& \lambda\int_{\mathcal{M}_{\Theta}}\frac{K(\sigma)F(w_n(\sigma))}{w_n(\sigma)^2}x_n(\sigma)^2d\sigma_g\\
                    &+& \lambda\int_{\mathcal{M}\setminus \mathcal{M}_{\Theta}}\frac{K(\sigma)F(w_n(\sigma))}{w_n(\sigma)^2}x_n(\sigma)^2d\sigma_g\nonumber
\end{eqnarray}
On the other hand from hypothesis $({\rm h}_{\infty, F})$ we have
$$
\lim_{|\xi|\rightarrow \infty}F(\xi)=+\infty.
$$
Hence, taking into account that $F\in C^0(\erre;\erre)$, there exists
a positive constant $\varrho_1$ such that
$$
F(\xi)\geq -\varrho_1,
$$
\noindent for every $\xi\in \erre$. Then, since $1/\|w_n\|_{H_\alpha^2}=x_n/w_n$, it follows that
\begin{eqnarray*}
 \int_{\mathcal{M}\setminus \mathcal{M}_{\Theta}}\frac{K(\sigma)F(w_n(\sigma))x_n(\sigma)^2}{w_n(\sigma)^2}d\sigma_g &\geq &
 -\varrho_1\frac{\|K\|_{\infty}}{\|w_n\|_{H_\alpha^2}^2}\textrm{Vol}_g(\mathcal{M}\setminus \mathcal{M}_{\Theta}).
\end{eqnarray*}

\noindent The above inequality together \eqref{felix2} and \eqref{deco} yield
  \begin{eqnarray*}
\frac{1}{2}-\frac{c+o(1)}{\|w_n\|^2_{H_\alpha^2}} &\geq & \lambda\essinf_{\sigma\in{{\mathcal{M}}}}K(\sigma)\int_{\mathcal{M}_{\Theta}}\frac{F(w_n(\sigma))}{w_n(\sigma)^2}x_n(\sigma)^2d\sigma_g  \nonumber\\
                    &-&\varrho_1\lambda\frac{\|K\|_{\infty}}{\|w_n\|_{H_\alpha^2}^2}\textrm{Vol}_g(\mathcal{M}\setminus \mathcal{M}_{\Theta}).\nonumber
\end{eqnarray*}
This is clearly impossible. Hence, the sequence $\{w_n\}$ is bounded in $X$.\par
\noindent In conclusion, applying the version of the mountain pass theorem, where (PS) is replaced by (C), the conclusion is achieved; see, for details, the work \cite{Cerami}.
\end{pf}
\begin{remark}\rm{
Condition \eqref{Gigio} is a consequence of the following
$$
\lim_{|t|\rightarrow \infty} \frac{\displaystyle f(t)}{t^2}=+\infty,
$$
that characterizes the problem $(P_\lambda)$ as superlinear at infinity. As pointed out in precedence the boundedness of Palais-Smale sequences of the Euler-Lagrange functional $J_\lambda$ can be obtained if the Ambrosetti-Rabinowitz condition is verified. However, hypothesis $(\rm AR)$ can not be useful treating some nonlinearities. Indeed, if a function $f$ satisfies $(\rm AR)$ then
there exist suitable (positive) constants $\gamma_1,\gamma_2$ with
$$
F(\xi)\geq \gamma_1 |\xi|^\mu-\gamma_2,\quad  \forall\; \xi\in\erre.
$$
For this reason, in recent years, several authors studied superlinear problems trying to drop the condition $(\rm AR)$; see for instance the works \cite{Liu,MS} and the references therein.}
\end{remark}

\begin{example} \label{Ex1Final}
{\rm
\noindent Set $f:\erre\rightarrow \erre$ be the continuous function given by
%2(abs(t)^3+1)
$$
f(t):=t\log(1+|t|)+1.
$$
Hence, we deduce that $$F(\xi)=\frac{|\xi|}{2}+\frac{(\xi^2-1)\log(|\xi|+1)}{2}-\frac{\xi^2}{4}+\xi,\,\,\,\,\forall\,\xi\in\erre$$ and it is easy to check that all the conditions of Theorem \ref{TheoremCerami} hold. Then, with the usual notations, the following equation
$$-\Delta_{g} w+\alpha (\sigma) w= \lambda K(\sigma)(w\log(1+|w|)+1), \eqno{(C_{\lambda})}$$
for every $\sigma\in {\mathcal{M}}$ and $w\in H_1^2({\mathcal{M}})$, admits at least two non-negative (and non-trivial) weak solutions for every $\lambda\in \Lambda_{\mathcal{M}}$. Finally, since
$$
\lim_{|\xi|\rightarrow +\infty}\frac{\xi f(\xi)}{F(\xi)}=\lim_{|\xi|\rightarrow +\infty}\frac{4\xi (\xi \log(|\xi|+1)+1)}{2|\xi|+2(\xi^2-1)\log(|\xi|+1)-\xi(\xi-4)}=2,
$$
the Ambrosetti-Rabinowitz hypothesis fails.
}
\end{example}

\section{Applications to singular elliptic problems}
\indent In this section $d$ and $s$ denote two real fixed constants with $d\geq 3$ and $1-d<s<0,$ $f:\erre\to\erre$ is a locally Lipschitz continuous function (or more generally locally H\"older continuous) and $K$ is a smooth and positive map on the unit sphere ${\mathbb{S}}^d$. Consider the following parameterized Emden-Fowler problem that arises in astrophysics, conformal Riemannian geometry, and in the theories of thermionic emission, isothermal stationary gas sphere, and gas combustion:
$$-\Delta u =
\lambda|x|^{s-2}K({x}/{|x|})f(|x|^{-s}u),\qquad x\in {\erre}
^{d+1}\setminus \{0\}.\eqno (\mathfrak{F}_\lambda)$$
As pointed out in Introduction, equation $(\mathfrak{F}_\lambda)$ has been
 studied by Cotsiolis-Iliopoulos
\cite{CI-BUll-Sci-math}, V\'azquez-V\'eron \cite{VV-C.R.Paris} by using either minimization or
minimax methods. More recently, in \cite{KR} and successively in \cite{BOMORa}, some existence results are achieved by variational methods. The solutions of $(\mathfrak{F}_\lambda)$ are being sought in the
particular form
\begin{equation}\label{transzformacio}
u(x)=r^sw(\sigma),
\end{equation}
where, $(r,\sigma):=(|x|,x/|x|)\in (0,\infty)\times {\mathbb{S}}^d$ are the spherical
coordinates in ${\erre}^{d+1}\setminus \{0\}$ and $w$ be a smooth function defined on ${\mathbb{S}}^d$.
 This type of
transformation is also used  by Bidaut-V\'eron and V\'eron
\cite{BV-V-Invent}, where the asymptotic of a special form of
$(\mathfrak{F}_\lambda)$ has been studied. Throughout
(\ref{transzformacio}), taking into account that
$$
\Delta u=r^{-d}\frac{\partial}{\partial r}\left(r^{d}\frac{\partial u}{\partial r}\right)+r^{-2}\Delta_{h}u,
$$
equation $(\mathfrak{F}_\lambda)$ reduces to
$$-\Delta_{h} w+ s(1-s-d)w= \lambda
K(\sigma)f(w),\qquad \sigma\in {\mathbb{S}}^d,\, w\in H_1^2({\mathbb{S}}^d).
$$
\indent Due to our regularity assumptions on the data, the weak solutions of our problem are also classical as observed by Krist\'aly and R\u adulescu in \cite{KR}. Now, for every two nonnegative constants $\gamma,\delta$, with $\gamma\neq\delta$, let
$$
a^{\star}_{\gamma}(\delta):= \frac{\displaystyle A^{\star}(\gamma)-qF(\delta)\|K\|_{L^1({\mathbb{S}}^d)}}{s(1-s-d)\omega_d(\gamma^2-\delta^2)q},
$$
\noindent where
$$
A^{\star}(\gamma):=(q(s(1-s-d)\omega_d)^{1/2}\gamma S_1a_1+(s(1-s-d)\omega_d)^{q/2}\gamma^{q}{S_q^qa_2})\|K\|_{\infty}.
$$

We have the following result.

\begin{corollary}\label{Cor2}
Let $f:\erre\to\erre$ is a locally Lipschitz continuous
function such that condition $({\rm h}_\infty)$ holds. Assume that there are three real
constants $\gamma_1,\gamma_2$ and $\delta$, with $0\leq \gamma_1 <
\delta< \gamma_2$, such that
\begin{equation}\label{EE12}
a^{\star}_{\gamma_2}(\delta)<a^{\star}_{\gamma_1}(\delta).
\end{equation}
\noindent Then, for each parameter $\lambda$ belonging to
$$
\Lambda^{\star}_{\mathfrak{F}}:=\left]\displaystyle \frac{1}{2a^{\star}_{\gamma_1}(\delta)},\frac{1}{2a^{\star}_{\gamma_2}(\delta)}\right[,
$$
the following problem
$$-\Delta u =
\lambda|x|^{s-2}K({x}/{|x|})f(|x|^{-s}u),\qquad x\in {\erre}
^{d+1}\setminus \{0\},\eqno (\mathfrak{F}_\lambda)$$
admits at least one classical solution.
\end{corollary}

\begin{pf} Let us choose
$(M,g)=({\mathbb{S}}^d, h),$ and $\alpha(\sigma):=s(1-s-d)$ for every
$\sigma\in {\mathbb{S}}^d$ in Theorem \ref{TT1}.
Clearly $\alpha\in C^{\infty}({\mathbb{S}}^{d})$ and, thanks to $1-d<s<0$, $\alpha$ to be positive on ${\mathbb{S}}^{d}$.
Thus, for every
$\Lambda^{\star}_{\mathfrak{F}}\subseteq \Lambda$, the problem
$$-\Delta_{h} w+ s(1-s-d)w= \lambda
K(\sigma)f(w),\qquad \sigma\in {\mathbb{S}}^d,\, w\in H_1^2({\mathbb{S}}^d),
$$
has at least one non-trivial
solution $w_\lambda\in H_1^2({\mathbb{S}}^d)$. On account of (\ref{transzformacio}), the element
$u_\lambda(x)=|x|^sw_\lambda(x/|x|),$ is a non-trivial
solution of $(\mathfrak{F}_\lambda).$
\end{pf}

A special case of Theorem \ref{CAR}, reads as follows.
\begin{corollary}\label{cork}
Let $f:\erre\to\erre$ be a locally Lipschitz continuous
function such that
conditions $({\rm h}_{\infty}')$ and $({\rm h}_{0,F}'')$ hold.
\noindent Then, there exists $\lambda^\star_{\mathfrak{F}}>0$ such that for every $\lambda\in ]0,\lambda^\star_{\mathfrak{F}}[$, the following problem
$$-\Delta u =
\lambda|x|^{s-2}K({x}/{|x|})f(|x|^{-s}u),\qquad x\in {\erre}
^{d+1}\setminus \{0\},\eqno (\mathfrak{F}_\lambda)$$
admits at least one non-trivial classical solution. Moreover if, in addition, the function $f$ satisfies
$$
    0<\mu \int_0^tf(s)ds\leq t f(t),
$$
for every $|t| \ge r$, for some $r>0$ and $\mu>2$, then, for each $\lambda\in]0,\lambda^\star_{\mathfrak{F}}[$, the problem $(\mathfrak{F}_\lambda)$
admits at least two non-trivial classical solutions.
\end{corollary}
\begin{remark}\label{maximalinterval2}
\rm{Arguing as in Corollary \ref{Cor2}, analogous of Theorems in Section 3 can be easily obtained for our new setting. For instance, the first part of Corollary \ref{cork} is an exhaustive version of Theorem \ref{ultimo} for Emden-Fowler type equations. In this case, for every parameter $\lambda\in ]0,\lambda^\star_{\mathfrak{F}}[$, the problem $(\mathfrak{F}_\lambda)$ admits at least one non-trivial solution. Moreover, the existence of a second non-trivial solution is obtained arguing as in Theorem \ref{CAR}. Moreover, by using Remarks \ref{maximalinterval}, a concrete expression for the value of $\lambda^\star_{\mathfrak{F}}$ is given by
$$
 \lambda^{\star}_{\mathfrak{F}}=
 \left\{
\begin{array}{l}
+\infty \quad\, {\rm if} \quad 1<q<2 \\
\displaystyle\frac{1}{\|K\|_{\infty}S^2_2a_2} \quad {\rm if} \quad q=2 \\
\displaystyle \displaystyle\frac{q(s(1-s-d)\omega_d)^{1/2}\overline{\gamma}_{\max}}{2\|K\|_{\infty}(qS_1a_1+(s(1-s-d)\omega_d)^{(q-1)/2}{S_q^qa_2}\overline{\gamma}_{\max}^{q-1})} \quad {\rm if} \quad q\in \left]2,\frac{2d}{d-2}\right[, \\
\end{array}
\right.
 $$
\noindent where
$$
\overline{\gamma}_{\max}:=\left(\frac{S_1a_1 }{(q-2)(s(1-s-d))^{q/2}\omega_d^{(q-1)/2}S_q^qa_2}\right)^{1/(q-1)}.
$$}
\end{remark}
\newpage
Finally, we give here a direct application of Corollary \ref{cork}.
\begin{example} \label{Ex225}
{\rm  Let $K\in C^{\infty}({\mathbb{S}}^{3})$ be a positive mapping and consider the parametric Emden-Fowler equation
$$-\Delta u =
\lambda |x|^{s-2}K({x}/{|x|})(1+|x|^{-3s}u^{3}),\qquad x\in {\erre}
^{4}\setminus \{0\},\eqno (\mathfrak{F}_\lambda')$$
\noindent where we fix $s\in]-2,0[$.
Then, for every $\lambda$ belonging to the interval
$$\Lambda_{\mathfrak{F}_\lambda'}:=\left]0, \frac{1}{3\|K\|_{\infty}S_1}\left(\frac{2}{S_4^4}\right)^{1/3}\right[,$$
the problem $(\mathfrak{F}_\lambda')$ admits at least two non-negative (and non-trivial) classical solutions.
To prove this, we can apply Corollary \ref{cork} to the locally Lipschitz continuous function
$$
f(t):=1+t^{3},\,\,\,\,\,\,\forall\, t\in\erre,
$$
bearing in mind Remark \ref{maximalinterval2}. Indeed, clearly the function $f$ satisfies \eqref{Su} and, since
$$
\lim_{t\rightarrow0^+} \frac{f(t)}{t}=+\infty,
$$
also condition $({\rm h}_{0,F}'')$ holds true. Moreover, taking into account that
$$
\lim_{|\xi|\rightarrow \infty}\frac{\xi f(\xi)}{F(\xi)}=4\lim_{|\xi|\rightarrow \infty}\frac{\xi^{3}+1}{\xi^{3}+4}=4>2,
$$
there exist $\mu>2$ and $r>0$ such that
$$
    0<\mu F(\xi)\leq \xi f(\xi),
    $$
    for every $|\xi|>r$. Hence, all the assumptions of Corollary \ref{CAR} are verified and the conclusion follows.
}
\end{example}
\begin{remark} \label{FinaleRemark}
{\rm It is well-known that sharp Sobolev inequalities are important in the study of partial differential
equations. In our context,
a concrete upper bound for the constants $S_q$ in the above example is essential for an explicit evaluation of the interval of parameters.
Now, we observe that in Example \ref{Ex225} a more precise information on the size of the interval $\Lambda_{\mathfrak{F}_\lambda'}$ can be easily obtained
taking into account Remark \ref{costanti}. Indeed, if $q\in [1,5]$ and $s\in ]-2,0[$, we have
$$
S_q^{\star}=\displaystyle
\frac{\omega_3^{\frac{2-q}{2q}}}{(-s(s+2))^{1/2}}.
$$
Consequently, for $\lambda$ sufficiently small,
more precisely
$$
0<\lambda<\displaystyle\frac{(-s(s+2))^{7/6}}{3}\left(\frac{2^{1/3}}{\omega_3^{1/6}\|K\|_{\infty}}\right),
$$
problem $(\mathfrak{F}_\lambda')$ admits at least two non-negative (and non-trivial) classical solutions.
}
\end{remark}
\medskip
 \indent {\bf Acknowledgements.}  V.~R\u adulescu acknowledges the support through  Grant CNCS PCE--47/2011.

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\end{document} 