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%\title[A nonlinear problem on compact...]
%{A nonlinear problem on compact\\ Riemmanian manifolds}
\title[Generalized Yamabe equations]
{Multiple solutions of generalized Yamabe equations on Riemmanian manifolds and applications to Emden--Fowler problems}
\author{Gabriele Bonanno, Giovanni Molica Bisci and Vicen\c{t}iu R\u{a}dulescu}
%

\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 P.A.U., Universit\`a  degli
Studi Mediterranea di Reggio Calabria, Salita Melissari - Feo di
Vito, 89100 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}

%\subjclass{Primary ; Secondary , }

\keywords{Yamabe problem, Emden-Fowler equation, sublinear eigenvalue problem, 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}[section]
\renewcommand{\theth}{\Alph{th}}
\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\NN{{\mathbb N} }
\def\ZZ{{\mathbb Z} }
\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 three nontrivial solutions for a nonlinear problem on compact $d$-dimensional ($d\geq 3$) Riemmanian manifolds without boundary, is established. This multiplicity result is then
applied to solve Emden-Fowler equations that involve sublinear
terms at infinity. Two concrete examples are also provided in the present paper.
Our results apply to problems arising in conformal Riemannian geometry, astrophysics, and in the theories of thermionic emission, isothermal stationary gas sphere, and gas combustion.
\end{abstract}
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\maketitle
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%INTRODUCTION%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Introduction}

\indent Analysis on Riemannian manifolds is a field currently undergoing great
development. More and more, analysis proves to be a very powerful tool
for solving geometrical problems. Conversely, geometry may help us to solve
certain problems in analysis, as pointed out in Aubin \cite{A-konyv}.

Let $({\mathcal{M}},g)$ be a compact $d$-dimensional  Riemannian manifold without boundary, where $d\geq 3$. Let
$\Delta_{g}$ denote the
Laplace-Beltrami operator on $({\mathcal{M}},g)$ and assume that the functions $\alpha, K\in C^{\infty}({\mathcal{M}})$
are positive. Suppose $f:\RR \to \RR$ is a locally H\"older continuous function with
{\it sublinear} growth and $\lambda$ is a positive real parameter.
In this paper we are interested in the existence of (classical) solutions to the following eigenvalue problem:
$$-\Delta_{g} w+\alpha (\sigma) w= \lambda
K(\sigma)f(w),\qquad \sigma\in {\mathcal{M}} ,\ w\in H_1^2({\mathcal{M}}).\eqno{(P_{\lambda})}$$

This problem generalizes the celebrated Yamabe equation (see Aubin \cite[p.~126]{aubin})
$$4\,\frac{d-1}{d-2}\, \Delta_g\varphi +R\varphi =\mu\, \varphi^{q-1}\qquad\mbox{in ${\mathcal{M}}$},$$
where $2<q<2d/(d-2)$ and $R$ denotes the scalar curvature of ${\mathcal{M}}$. According to Berger \cite{berger}, curvature is ``the No.~1 Riemannian invariant and the most natural. Gauss and then Riemann saw it instantly". The main question in the fundamental Yamabe's paper \cite{yamabe} was whether there are any restrictions needed to have a metric of constant scalar curvature. Yamabe proved that for compact manifolds there always exists such a metric conformally equivalent to any given metric on the manifold. This proof turned out to be incomplete and Aubin \cite{aubinjmpa} proved the theorem for all manifolds of dimension $d\geq 6$ that were not conformally flat. To settle the problem completely, a new global type of argument was needed, and that was provided in 1984 by Schoen \cite{schoen}.

By using variational methods (see Theorem \ref{abstract} below), we find a well determined open interval of values of the parameter $\lambda$ for which problem $(P_{\lambda})$ admits at least three
nontrivial solutions. It is worth noticing that, at best of our knowledge, this is the first result in which all the three solutions are nontrivial.

 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 $({\mathbb{S}}^d,h)$.

 Indeed, existence results for problem $(S_\lambda)$ yield, by using an appropriate change of coordinates, the existence of solutions to the following parameterized Emden-Fowler equation $$-\Delta u =
\lambda|x|^{s-2}K({x}/{|x|})f(|x|^{-s}u),\qquad x\in \RR
^{d+1}\setminus \{0\},\eqno(\mathfrak{F}_\lambda)$$
see Remark \ref{VV} and Corollary \ref{Cor2}.

 Moreover, 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 famous 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-C.R.Acad,CI-BUll-Sci-math}, 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.

  Cotsiolis and Iliopoulos \cite{CI-BUll-Sci-math} and V\'azquez and  V\'eron \cite{VV-C.R.Paris} studied problem $(\mathfrak{F}_\lambda)$, by applying either minimization or
minimax methods, provided that $f(t)=|t|^{p-1}t$, with $p>1$. Successively, 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.
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
Further, 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}.

In the present paper we use a new approach to attach sublinear problems at infinity, previously developed in Bonanno and Molica Bisci \cite{BOMO}. We obtain the existence of a well localized open interval of positive parameters for which problem $(P_{\lambda})$ admits at least three nontrivial solutions; see Theorem \ref{Main1} and Remark \ref{f0}.

 The present paper is organized as follows. In Section 2 we recall some basic definitions
and preliminary facts on the Sobolev
spaces defined on compact Riemannian manifolds, while Section 3 is devoted to the existence of at least three solutions for the eigenvalue
problem $(P_{\lambda})$. In Section 4 we give some consequences of the main results, as well as the existence of three nontrivial solutions for Emden-Fowler equations.
 A concrete example of application of our main theorems is then presented in the last section. We cite the very 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.

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 $\alpha\in C^\infty({\mathcal{M}})$ be a positive
function and put $\|\alpha\|_\infty:=\displaystyle\max_{\sigma\in {\mathcal{M}}} \alpha(\sigma)$. For every $w\in C^\infty ({\mathcal{M}}),$ set
$$\|w\|_{H_\alpha^2}^2:=\int_{\mathcal{M}} |\nabla w(\sigma)|^2 d\sigma_g+\int_{\mathcal{M}}\alpha(\sigma)|w(\sigma)|^2 d\sigma_g,$$ 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 symbols and $g^{lk}$ are the elements of the inverse matrix of $g$.\par
 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
$$\langle w_1,w_2\rangle_{H_\alpha^2}=\int_{\mathcal{M}}\langle
\nabla w_1,\nabla w_2 \rangle_g
d\sigma_g+\int_{\mathcal{M}}\alpha(\sigma)\langle w_1,w_2 \rangle_g
d\sigma_g, \qquad w_1,w_2\in H_\alpha ^2({\mathcal{M}}),$$
where $\langle \cdot,\cdot \rangle_g$ is the
inner product on covariant tensor fields associated to $g.$\par
 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,{\min_{\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 From the Rellich-Kondrachov theorem for compact manifolds without boundary one has
$$
H_\alpha ^2({\mathcal{M}})\hookrightarrow L^q({\mathcal{M}}),
$$
for every $q\in [1,2d/(d-2)].$ In particular, the embedding is compact whenever $q\in [1,2d/(d-2))$.\par
\noindent Hence, there exists a positive constant $S_q$ such that
\begin{equation}\label{ineq}
\|w\|_q\leq S_q \|w\|_{H_\alpha^2}, \qquad\mbox{for all}\; w\in H_\alpha^2({\mathcal{M}}).
\end{equation}

 From now on, we assume that the nonlinearity $f$ satisfies the following structural condition:\par
{\it
\indent $f:\R\to \R$ is a locally H\"older continuous function
sublinear at infinity, that is,}
$$
\lim_{|t|\to\infty}\frac{f(t)}{t}=0.\leqno({\rm h}_\infty)
$$
\noindent Let $K\in C^{\infty}({\mathcal{M}})$ be a positive function.

 We recall that a function $w\in H_1^2({\mathcal{M}})$ is a {\it{weak solution}} of problem $(P_{\lambda})$ if
$$
\int_{\mathcal{M}} \langle
\nabla w,\nabla v \rangle_g
d\sigma_g+\int_{\mathcal{M}}\alpha(\sigma)\langle w ,v \rangle_g
d\sigma_g-\lambda\int_{\mathcal{M}} K(\sigma)f(w(\sigma))v(\sigma)d\sigma_g=0,\quad
$$
\noindent for every $v\in H_1^2({\mathcal{M}}).$\par
Further, due to the regularity assumption on $f$, the weak solutions are classical.
\noindent Hence, the main purpose is to study the following problem:\par
\smallskip
\noindent {\it Find $\lambda >0$ and $w\in H_1^2({\mathcal{M}})$ such
that}
$$-\Delta_{g}w+\alpha (\sigma)w= \lambda
K(\sigma)f(w),\qquad \mbox{for all}\ \sigma\in {\mathcal{M}},\ w\in H_1^2({\mathcal{M}}). \eqno{(P_{\lambda})}$$
\noindent Here, $\Delta_{g}$ represents the Laplace-Beltrami operator that, applied to a function $w\in H_1^2({\mathcal{M}})$, is given (locally) by the following expression
$$
\Delta_g w=g^{ij}\left(\frac{\partial^2w}{\partial x^i\partial x^j}-\Gamma^{k}_{ij}\frac{\partial w}{\partial x^k}\right).
$$
\begin{remark}
\rm{For a fixed $\lambda>0$, the function
$w_\lambda(\sigma)=c\in \RR\setminus\{0\} $, is a solution of $(P_\lambda)$
if and only if the function $\sigma\mapsto
{\lambda K(\sigma)}/{\alpha(\sigma)}$ is constant. In this case, nontrivial constant
solutions of $(P_\lambda)$, appear as fixed points of the function
$t\mapsto k_\lambda f(t),$ where $\kappa_\lambda$ denotes the constant value ${\lambda K(\sigma)}/{\alpha(\sigma)}$.}
\end{remark}

In order to obtain multiple solutions of $(P_\lambda)$ not only
in the case of constant solutions, we  use variational methods.
The main tool is a critical point theorem that we recall here in a convenient form. This result has been obtained in Bonanno and Marano \cite{BONMAR} and it is a more precise version of Theorem 3.2 of Bonanno and Candito \cite{BONCAN}.

\begin{theorem}\label{abstract}
 Let $X$ be a reflexive real Banach space, $\Phi:X\to\erre$ be a coercive, continuously G$\hat{a}$teaux differentiable and sequentially weakly lower semi-continuous functional whose G$\hat{a}$teaux derivative admits a continuous inverse on $X^*$, $\Psi :X\to \R$ be a continuously G$\hat{a}$teaux differentiable functional whose $G\hat{a}$teaux derivative is compact such that $$ \Phi (0)=\Psi (0)=0.$$
Assume that there exist $r>0$ and $\bar{x}\in X$, with $r<\Phi(\bar{x})$, such that:
\begin{itemize}
\item[$(a_1)$] $\dfrac{\displaystyle\sup_{x\in\Phi^{-1}(]-\infty,r])}\Psi(x)}{r}< \frac{\Psi(\bar{x})}{\Phi(\xseg)}$$;$
\item[$(a_2)$] For each $\lambda\in \Lambda_{r}:=\left]\displaystyle\frac{\Phi(\xseg)}{\Psi(\bar{x})},\frac{r}{\displaystyle\sup_{x\in\Phi^{-1}(]-\infty,r])}\Psi(x)}\right[$ the functional $J_\lambda:=\Phi -\lambda \Psi$ is coercive.
\end{itemize}
Then, for each $\lambda \in \Lambda_{r}$, the functional $J_\lambda$ has at least three distinct critical points in $X$.
\end{theorem}
We recall that
the derivative of $\Phi$ admits a continuous inverse on $X^*$ when %we mean that
there exists a continuous operator $T:X^*\rightarrow X$ such that $T(\Phi'(x))=x$ for all $x\in X$.
\section{Main results}

\indent We set
$$
\kappa_\alpha:=\Big(\frac{2}{\|\alpha\|_{L^1({\mathcal{M}})}}\Big)^{1/2},
$$
\noindent and
$$
K_1:=\frac{S_1}{\sqrt{2}}\,\|\alpha\|_{L^1({\mathcal{M}})},\,\,\,\,\,\,\,\,
K_2:=\frac{S_q^q}{2^{\frac{2-q}{2}}q}\,\|\alpha\|_{L^1({\mathcal{M}})}.
$$
\noindent Further, let
$$F(\xi):=\int_0^{\xi}f(t)\;dt,$$
\noindent for every $\xi\in\R$.

 \smallskip
 The main abstract theorem in this paper is the following multiplicity result.

\begin{theorem} \label{Main1}
Let $f:\RR \to\RR $ be a function such that $(\rm{h}_{\infty})$ holds and assume that
\begin{itemize}
\item [$(\textrm{h}_1)$] There exist two nonnegative constants $a_1,a_2$ such that
$$
|f(t)|\leq a_1+a_2|t|^{q-1},\qquad\mbox{for all}\ t\in\erre,
$$
where $q\in ]1,
2d/(d-2)[$$;$
\item [$(\textrm{h}_2)$] There exist two positive constants $\gamma$ and $\delta$, with $\delta>\gamma\kappa_\alpha$, such that
$$
\frac{\displaystyle F(\delta)}{\delta^2}>\frac{\|K\|_\infty}{\|K\|_{L^1({\mathcal{M}})}}\left(a_1\frac{K_1}{\gamma}+a_2K_2\gamma^{q-2}\right).
$$
\end{itemize}

\noindent Then, for each parameter $\lambda$ belonging to
$$
\Lambda_{(\gamma,\delta)}:=\left]\frac{\delta^2
\|\alpha\|_{L^1({\mathcal{M}})}}{2F(\delta)\|K\|_{L^1({\mathcal{M}})}},\frac{\|\alpha\|_{L^1({\mathcal{M}})}}{2\|K\|_\infty\Big(a_1\displaystyle\frac{K_1}{\gamma}+a_2K_2\gamma^{q-2}\Big)}\right[,
$$
\noindent the problem
$(P_\lambda)$ possesses at least three distinct solutions in $H_1^2({\mathcal{M}})$.
\end{theorem}
\begin{pf}
Our aim is to apply Theorem \ref{abstract}. Hence, let $X:=H_1 ^2({\mathcal{M}})$ and consider the
functionals $\Phi,\Psi:X\to\R$ 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 $\Phi:X\to\R$ is a coercive, continuously G\^{a}teaux differentiable and sequentially weakly lower semi-continuous functional whose G\^{a}teaux derivative admits a continuous inverse on $X^*$. On the other
hand, $\Psi$ is
well-defined, continuously G\^{a}teaux differentiable and with
compact derivative. Moreover, one has
$$
\Phi'(w)(v)=\int_{\mathcal{M}} \langle
\nabla w,\nabla v \rangle_g
d\sigma_g+\int_{\mathcal{M}} \alpha(\sigma)\langle w(\sigma),v(\sigma)\rangle_g d\sigma_g,$$
\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 weak solutions (hence classical solutions) of problem $(P_\lambda)$. Now, $\Phi (0)=\Psi (0)=0$ and
since condition $(\textrm{h}_1)$ holds, one has that
\begin{equation}\label{inequality}
F(\xi)\leq a_1|\xi|+a_2\frac{|\xi|^{q}}{q},
\end{equation}
for every $\xi\in\erre$.\par
\noindent Let $\varrho\in ]0,+\infty[$ and consider the function
$$
\chi(\varrho):=\frac{\sup_{w\in\Phi^{-1}(]-\infty,\varrho])}\Psi(w)}{\varrho}.
$$
\noindent Taking into account (\ref{inequality}), 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 $w\in\Phi^{-1}(]-\infty,\varrho])$, owing to (\ref{ineq}), we get
$$
\Psi(w)\leq \|K\|_\infty\Big((2\varrho)^{1/2}S_1a_1+\frac{2^{q/2}S_q^qa_2}{q}\varrho^{q/2}\Big).
$$
Hence, by using the definition of $\Phi$, one has
\begin{eqnarray}\label{hh}
\sup_{w\in\Phi^{-1}(]-\infty,\varrho])}\Psi(w)\leq \|K\|_\infty\left((2\varrho)^{1/2}S_1a_1+\frac{2^{q/2}S_q^qa_2}{q}\varrho^{q/2}\right).
\end{eqnarray}
\noindent From (\ref{hh}), the following inequality holds
\begin{equation}\label{n}
\chi(\varrho)\leq \|K\|_\infty\left(\sqrt{\frac{2}{\varrho}}S_1a_1+\frac{2^{q/2}S_q^qa_2}{q}\varrho^{q/2-1}\right)
\end{equation}
\noindent for every $r>0$.\par
\noindent Next, put $w_\delta(\sigma):=\delta$ for every $\sigma\in M$. 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)\nonumber \\
               & & =\frac{1}{2}\int_{\mathcal{M}} \alpha(\sigma)\delta^2\;d\sigma_g=\frac{\delta^2}{2}\|\alpha\|_{L^1({\mathcal{M}})}.
\end{eqnarray}
Taking into account that $\delta>\gamma\kappa_\alpha$, by a direct computation,
one has
$\gamma^2<\Phi(w_\delta)$.


\noindent Moreover,
\begin{eqnarray}\label{l}
\Psi(w_\delta)=\int_{\mathcal{M}} K(\sigma)F(w_\delta(\sigma))\;d\sigma_g=F(\delta)\|K\|_{L^1({\mathcal{M}})}.
\end{eqnarray}
\noindent Hence, from (\ref{norm}) and (\ref{l}), one has
\begin{equation}\label{condition}
\frac{\Psi(w_\delta)}{\Phi(w_\delta)}=2\frac{\displaystyle F(\delta)\|K\|_{L^1({\mathcal{M}})}}{\delta^2\|\alpha\|_{L^1({\mathcal{M}})}}.
\end{equation}
In view of $(\textrm{h}_2)$ and taking into account (\ref{n}) and (\ref{condition}), we get
\begin{eqnarray}%\label{c}
\chi(\gamma^2)=\frac{\displaystyle\sup_{w\in\Phi^{-1}(]-\infty,\gamma^2])}\Psi(w)}{\gamma^2}
& & \leq \|K\|_\infty\Big(\frac{\sqrt{2}}{\gamma}S_1a_1+\frac{2^{q/2}S_q^qa_2}{q}\gamma^{q-2}\Big)\nonumber \\
& & =\frac{2 \|K\|_\infty}{\|\alpha\|_{L^1({\mathcal{M}})}}\Big(a_1\displaystyle\frac{K_1}{\gamma}+a_2K_2\gamma^{q-2}\Big)\nonumber\\
               & & <2\frac{\displaystyle F(\delta)\|K\|_{L^1({\mathcal{M}})}}{\delta^2\|\alpha\|_{L^1({\mathcal{M}})}}\nonumber \\
               & & =\frac{\Psi(w_\delta)}{\Phi(w_\delta)}.
               \nonumber
\end{eqnarray}
\noindent Therefore, the assumption $(a_1)$ of Theorem \ref{abstract} is satisfied taking $\bar{x}:=w_\delta$ and by choosing $r:=\gamma^2$. Moreover, owing to (${\rm h}_\infty$), for every
 $\varepsilon >0$ sufficiently small there is $c(\varepsilon)>0$ such that $|f(t)|\leq
\varepsilon|t|+c(\varepsilon)$
 for every $t\in \RR .$ Consequently, for every $w\in
 H_1^2({\mathcal{M}}),$ we have $$
J_\lambda(w)\geq\frac{1}{2}(1-\lambda\varepsilon\|
K\|_\infty
S_2^2)\|w\|_{H_\alpha^2}^2-c(\varepsilon)\lambda\|
K\|_\infty S_1\|w\|_{H_\alpha^2}. $$
Therefore, the  functional $J_\lambda$
 is coercive for every positive parameter, in particular, for every
 $$\lambda\in \Lambda_{(\gamma,\delta)}\subseteq \left]\displaystyle\frac{\Phi(w_\delta)}{\Psi(w_\delta)},\displaystyle\frac{\gamma^2}{\di\sup_{w\in\Phi^{-1}(]-\infty,\gamma^2])}\Psi(w)}\right[.$$

  \noindent So, condition $(a_2)$ holds and hence, all the assumptions of Theorem \ref{abstract} are satisfied. Then, for each $\lambda \in \Lambda_{(\gamma,\delta)}$, the functional $J_\lambda$ has at least three distinct critical points that are classical solutions of problem $(P_\lambda)$.
\end{pf}

\begin{remark}\label{GB}\rm{
\noindent Hypothesis $(\textrm{h}_\infty)$ can be substituted by the following growth condition:
\begin{itemize}
\item [$(\textrm{h}_\infty')$]
\textit{There exist two positive constants $b$ and $s<2$ such that
$$
F(\xi)\leq b(1+|\xi|^s),
$$
\noindent for every $\xi\in\erre$.}
\end{itemize}
 Indeed, owing to $(\textrm{h}_1)$, problem $(P_\lambda)$ is well defined. Therefore, the functional $J_\lambda$ is coercive for every $\lambda\in (0,\infty)$. Indeed, fixing $\lambda>0$, since $s<2$,
from the H\"{o}lder's inequality we have
$$
\int_{{\mathcal{M}}}|w(\sigma)|^sd\sigma_g\leq \|w\|_{L^2({\mathcal{M}})}^{s}{{\meas}_g({\mathcal{M}})}^{\frac{2-s}{2}},\qquad\mbox{for all}\ w\in {H_1^2({\mathcal{M}})}.
$$
\noindent Now, bearing in mind (\ref{ineq}), we obtain
\begin{equation}\label{co}
\int_{{\mathcal{M}}}|w(\sigma)|^sd\sigma_g\leq S_2^s\|w\|^{s}_{H_\alpha^2}{{\meas}_g({\mathcal{M}})}^{\frac{2-s}{2}},\qquad\mbox{for all}\ w\in {H_1^2({\mathcal{M}})}.
\end{equation}
So, by using (\ref{co}) and
from condition $(\textrm{h}_\infty')$, it follows that
$$
J_\lambda(w)\geq \frac{\|w\|^2_{H_\alpha^2}}{2}-\lambda b{\meas}_g({\mathcal{M}})^{\frac{2-s}{2}}S_2^s\|w\|^{s}_{H_\alpha^2}-\lambda b{\meas}_g({\mathcal{M}}),\quad\mbox{for all}\ w\in {H_1^2({\mathcal{M}})}.
$$
\noindent Hence, $J_\lambda$ is coercive for every real positive parameter $\lambda$.}
\end{remark}

\begin{remark}\label{f0}
\rm{Under the additional hypothesis $f(0)\neq 0$, Theorem \ref{Main1} ensures the existence of at least three nontrivial solutions. Indeed, in this case, zero is not a solution for the problem $(P_{\lambda})$, as a simple computation shows. Hence, all the three solutions, attained by using our abstract framework, are nontrivial. Moreover, if $f$ is only continuous instead of H\"older continuous, our result guarantees the existence of at least three weak (nontrivial) solutions for problem $(P_{\lambda})$.}
\end{remark}
\begin{remark}
\rm{The technical approach used to prove the previous result have been introduced in Bonanno and Molica Bisci \cite{BOMO}. In the cited work, the existence of at least three weak solutions for a Dirichlet problem is showed under suitable conditions on the potential $F$.}
\end{remark}

\begin{remark}
\rm{ Let $f:\R\to\R$ be a continuous and positive function in $\R\setminus\{0\}$ such that
$$
|f(t)|\leq a_2|t|^{q-1},\quad\forall\; t\in\erre,
$$
for some $q\in ]2,
2d/(d-2)[$.\par
 \noindent Clearly, the above growth condition is a particular case of hypothesis $(\textrm{h}_1)$ and implies $f(0)=0$. In this setting, under the additional hypothesis $(\textrm{h}_\infty')$, Theorem \ref{Main1} ensures the existence of at least three (two nontrivial) solutions for every
$$
\lambda>\lambda^{*}:=\frac{1}{2}\frac{\|\alpha\|_{L^1({\mathcal{M}})}}{\|K\|_{L^1({\mathcal{M}})}}\inf_{\delta>0}\frac{\delta^2}{F(\delta)}.
$$
This particular result can be achieved by using \cite[Theorem 9.1]{KRV}.
}
\end{remark}


\begin{remark}\label{costanti}
\rm{It is well-known that sharp Sobolev inequalities are important in the study of partial differential
equations, especially in the study of those arising from geometry and physics. There has
been much work on such inequalities and their applications. In our context,
a concrete upper bound for the constants $S_q$ in Theorem \ref{Main1} is essential for a concrete evaluation of the interval $\Lambda_{(\gamma,\delta)}$.
  In the particular case $({\mathcal{M}},g)=({\mathbb{S}}^{d},h)$, if $q\in[1,2d/(d-2)[$, one has
\begin{equation}\label{inequality2}
\displaystyle S_q\leq\frac{\kappa_q}{\min\left\{1,\displaystyle\min_{\sigma\in {\mathbb{S}}^{d}}\alpha(\sigma)^{1/2}\right\}},
\end{equation}
\noindent where, we set
\begin{equation*}\label{K}
\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 $w\in H^{2}_1({\mathbb{S}}^{d})$, one has
$$
\left(\int_{{\mathbb{S}}^{d}}|w(\sigma)|^qd\sigma_h\right)^{2/q}\leq \frac{q-2}{d\omega_d^{1-2/q}}\int_{{\mathbb{S}}^{d}}|\nabla w(\sigma)|^2d\sigma_h+\frac{1}{\omega_d^{1-2/q}}\int_{{\mathbb{S}}^{d}}|w(\sigma)|^2d\sigma_h,
$$
\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 $w\in H^{2}_1({\mathbb{S}}^{d})$.
}
\end{remark}

\section{Applications to nonlinear eigenvalue problems and Emden-Fowler equations}

\indent Let $\alpha, K\in C^{\infty}({\mathbb{S}}^{d})$
be positive and set
\begin{equation}\label{q1}
K_1^{\star}:=\frac{\kappa_1\|\alpha\|_{L^1({\mathbb{S}}^{d})}}{\sqrt{2}\min\left\{1,\displaystyle\min_{\sigma\in {\mathbb{S}}^{d}}\alpha(\sigma)^{1/2}\right\}}.
\end{equation}
\noindent Further, for $q\in ]1,2d/(d-2)[$, we will denote
\begin{equation}\label{b2}
K_2^{\star}:=
\frac{\kappa_q^q\|\alpha\|_{L^1({\mathbb{S}}^d)}}{2^{\frac{2-q}{2}}q\min\left\{1,\displaystyle\min_{\sigma\in {\mathbb{S}}^{d}}\alpha(\sigma)^{q/2}\right\}}.
\end{equation}
\noindent As a consequence of Theorem \ref{Main1}, and taking into account Remark \ref{costanti}, we get the following result on the existence of three solutions for nonlinear eigenvalues problems on the unit sphere ${\mathbb{S}}^d$.
\begin{corollary} \label{Cor1}
 Let $f:\RR \to\RR $ be a function such that $(\rm{h}_\infty)$ and $(\rm{h}_1)$ hold. Further, assume that there exist two positive constants $\gamma$ and $\delta$, with $\delta>\gamma\kappa_\alpha$, and
 \begin{itemize}
\item [$(\textrm{h}_2^{\star})$] $\displaystyle\frac{\displaystyle F(\delta)}{\delta^2}>\frac{\|K\|_\infty}{\|K\|_{L^1({\mathbb{S}}^d)}}\left(a_1\frac{K_1^{\star}}{\gamma}+a_2K_{2}^{\star}\gamma^{q-2}\right)$,\par
\end{itemize}
\noindent where $K_1^{\star}$ and $K_2^{\star}$ are given respectively by (\ref{q1}) and (\ref{b2}).\par
\noindent Then, for each parameter $\lambda$ belonging to
$$
\Lambda_{(\gamma,\delta)}^{\star}:=\left]\frac{\delta^2\|\alpha\|_{L^1({\mathbb{S}}^d)}}{2F(\delta)\|K\|_{L^1({\mathbb{S}}^d)}},\frac{\|\alpha\|_{L^1({\mathbb{S}}^d)}}{2\|K\|_\infty\Big(a_1\displaystyle\frac{K_1^{\star}}{\gamma}+a_2K_{2}^{\star}\gamma^{q-2}\Big)}\right[,
$$
\noindent the problem
$$-\Delta_{h} w+\alpha (\sigma) w= \lambda
K(\sigma)f(\omega),\qquad \sigma\in {\mathbb{S}}^d,\ w\in H_1^2({\mathbb{S}}^d),\eqno{(S_\lambda^{\alpha})}$$
 possesses at least three distinct solutions.
\end{corollary}

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

\indent Next, we 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:
$$-\Delta u =
\lambda|x|^{s-2}K({x}/{|x|})f(|x|^{-s}u),\qquad x\in \RR
^{d+1}\setminus \{0\}.\eqno (\mathfrak{F}_\lambda)$$
The equation $(\mathfrak{F}_\lambda)$ has been
 studied when $f$ has the form
$f(t)=|t|^{p-1}t,$ $p>1,$ see Cotsiolis-Iliopoulos
\cite{CI-BUll-Sci-math}, V\'azquez-V\'eron \cite{VV-C.R.Paris}. In
these papers, the authors obtained existence and multiplicity results for $(\mathfrak{F}_\lambda)$, applying either minimization or
minimax methods.

\begin{remark}\label{VV}
\rm{
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 $\RR ^{d+1}\setminus \{0\}$ 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
$$
\Delta u=r^{-d}\frac{\partial}{\partial r}\left(r^{d}\frac{\partial u}{\partial r}\right)+r^{-2}\Delta_{h}u,
$$
the equation $(\mathfrak{F}_\lambda)$ reduces to
$$-\Delta_{h} w+ s(1-s-d)w= \lambda
K(\sigma)f(w),\qquad \sigma\in {\mathbb{S}}^d,\, w\in H_1^2({\mathbb{S}}^d),
$$}
\noindent see also Krist\'aly and R\u adulescu \cite{KR}.
\end{remark}
From Remark \ref{VV}, we have the following result.\par
\begin{corollary}\label{Cor2}
Assume that  $d$ and $s$ are two constants such that $1-d<s<0.$ Further, let $K\in C^\infty({\mathbb{S}}^d)$ be a positive
function and $f:\RR \to \RR $ as in
 Corollary \ref{Cor1}. Then, for each parameter $\lambda$ belonging to
$$
\Lambda_{(\gamma,\delta)}^{s,d}:=\left]\frac{s(1-s-d)\omega_d\delta^2}{2F(\delta)\|K\|_{L^1({\mathbb{S}}^d)}},\frac{s(1-s-d)\omega_d}{2\|K\|_\infty\Big(a_1\displaystyle\frac{K_1^{\star}}{\gamma}+a_2K_{2}^{\star}\gamma^{q-2}\Big)}\right[,
$$
the following problem
$$-\Delta u =
\lambda|x|^{s-2}K({x}/{|x|})f(|x|^{-s}u),\qquad x\in \RR
^{d+1}\setminus \{0\},\eqno (\mathfrak{F}_\lambda)$$
admits at least three distinct solutions.
\end{corollary}

\noindent {\it Proof of Corollary \ref{Cor2}.} Let us choose
$(M,g)=({\mathbb{S}}^d, h),$ and $\alpha(\sigma):=s(1-s-d)$ for every
$\sigma\in {\mathbb{S}}^d$ in Corollary \ref{Cor1}.
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
$\lambda\in\Lambda_{(\gamma,\delta)}^{s,d}$, 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 three distinct
solutions $w_\lambda^i\in H_1^2({\mathbb{S}}^d),$ $i\in\{1,2,3\}$. On account of (\ref{transzformacio}), the elements
$u_\lambda^i(x)=|x|^sw_\lambda^i(x/|x|),$ $i\in \{1,2,3\},$ are
solutions of $(\mathfrak{F}_\lambda).$\par

\section{Some Examples}

In the next example, problem $(P_{\lambda})$ admits three nontrivial solutions owing to Theorem \ref{Main1}, while \cite[Theorem 1.1]{KR} as well as \cite[Theorem 9.4, p. 222]{KRV} cannot be applied.

\begin{example}\label{esempio1}
\rm{
Let $({\mathcal{M}},g)$ be a compact $d$-dimensional ($d\geq 3$) Riemannian manifold without boundary, fix $q\in ]2,2d/(d-2)[$ and let $K\in C^{\infty}({\mathcal{M}})$ be a positive function. Moreover, let $h:\erre\rightarrow\erre$ be the locally Lipschitz continuous function defined by
\[
h(t):= \left\{
\begin{array}{ll}
\displaystyle 1+|t|^{q-1} & \mbox{ if\, $|t|\leq r$} \\\\
\displaystyle \frac{(1+r^{2})(1+r^{q-1})}{1+t^2} & \mbox{ if $|t|> r$},
\end{array}
\right.
\]
\noindent where $r$ is a fixed constant such that
\begin{eqnarray}\label{ineq3}
r>\max\left\{\Big(\frac{2}{\meas_g({\mathcal{M}})}\Big)^{1/2},q^{\frac{1}{q-2}}\Big(\frac{\|K\|_\infty}{\|K\|_{L^1({\mathcal{M}})}}(K_1+K_2)\Big)^{\frac{1}{q-2}}\right\}.
\end{eqnarray}
\noindent  Clearly $h(0)\neq 0$ and $h(t)\leq (1+|t|^{q-1})$ for every $t\in\erre$. Hence, the condition $(\rm{h}_1)$ is satisfied for $a_1=a_2=1$. Moreover, one has that $\displaystyle\lim_{|t|\rightarrow \infty}h(t)/t=0$. Finally, owing to (\ref{ineq3}), it follows that
$$
\frac{\|K\|_\infty}{\|K\|_{L^1({\mathcal{M}})}}(K_1+K_2)<\frac{r^{q-2}}{q}.
$$
Therefore,
$$
\frac{\displaystyle\int_0^rh(t)dt}{r^2}=\frac{r^{q-2}}{q}+\frac{1}{r}>\frac{\|K\|_\infty}{\|K\|_{L^1({\mathcal{M}})}}(K_1+K_2),
$$
and condition $(\textrm{h}_2)$ holds choosing $\delta=r$.\par
\noindent Consequently, from Theorem \ref{Main1}, \noindent for each parameter
$$
\lambda\in\left]\frac{qr^2\meas_g({\mathcal{M}})}{2(qr+r^q)\|K\|_{L^1({\mathcal{M}})}},\frac{\meas_g({\mathcal{M}})}{2\|K\|_\infty(K_1+K_2)}\right[,
$$ the following problem
$$-\Delta_{g}w+w= \lambda
K(\sigma)h(w),\qquad \sigma\in {\mathcal{M}},\ w\in H_1^2({\mathcal{M}}),$$
 possesses at least three nontrivial solutions.\par
}\end{example}

The next example follows
directly by Corollary \ref{Cor1}. Moreover, the existence of at least three nontrivial solution for a class of Emden-Fowler equations is achieved.

\begin{example}\label{esempio2}
\rm{

 Consider three positive constants $a_1,a_2,\epsilon$, fix $q\in ]2,2d/(d-2)[$, where $d\geq 3$, and let $\alpha, K\in C^{\infty}({\mathbb{S}}^{d})$
be two positive functions.
Define $g:\erre\rightarrow\erre$ as follows
\[
g(t):= \left\{
\begin{array}{ll}
\displaystyle a_1+a_2|t|^{q-1} & \mbox{ if\, $|t|\leq \max\left\{\kappa_\alpha,\rho\right\}+\epsilon$} \\\\
\displaystyle a_1+a_2(\max\left\{\kappa_\alpha,\rho\right\}+\epsilon)^{q-1} & \mbox{ if $|t|> \max\left\{\kappa_\alpha,\rho\right\}+\epsilon$},
\end{array}
\right.
\]
\noindent where
$$
\kappa_\alpha:=\Big(\frac{2}{\|\alpha\|_{L^1({\mathbb{S}}^{d})}}\Big)^{1/2},
$$
 \noindent and
$$
\di\rho:=\left(\frac{\|K\|_\infty\|\alpha\|_{L^1({\mathbb{S}}^{d})}}{\|K\|_{L^1({\mathbb{S}}^{d})}}\right)^{\frac{1}{q-2}}
\left(\frac{qa_1\kappa_1^1}{\sqrt{2}\min\left\{1,\displaystyle\min_{\sigma\in {\mathbb{S}}^{d}}\alpha(\sigma)^{1/2}\right\}}+\frac{2^{\frac{q-2}{2}}a_2\kappa_q^q}{\min\left\{1,\displaystyle\min_{\sigma\in {\mathbb{S}}^{d}}\alpha(\sigma)^{q/2}\right\}}\displaystyle\right)^{\frac{1}{q-2}}.
$$
\noindent From Corollary \ref{Cor1}, \noindent for each parameter

$$
\lambda\in\left]\frac{q(\max\left\{\kappa_\alpha,\rho\right\}+\epsilon)^2\|\alpha\|_{L^1({\mathbb{S}}^d)}}{2(qa_1(\max\left\{\kappa_\alpha,\rho\right\}+\epsilon)+a_2(\max\left\{\kappa_\alpha,\rho\right\}+\epsilon)^q)\|K\|_{L^1({\mathbb{S}}^d)}},\frac{q\|\alpha\|_{L^1({\mathbb{S}}^d)}}{
2\|K\|_{L^1({\mathbb{S}}^d)}\rho^{{q-2}}}\right[,
$$

\noindent the following problem
$$-\Delta_{h}w+\alpha (\sigma)w= \lambda
K(\sigma)g(w),\ \sigma\in {\mathbb{S}}^d,\ w\in H_1^2({\mathbb{S}}^d),$$
 possesses at least three nontrivial solutions.\par
 \noindent Finally, let $s$ be a constant such that $d-1<s<0$. As consequence of Corollary \ref{Cor2}, we obtain that, for every
$$
\lambda\in\left]\frac{sq(1-s-d)(\max\left\{\kappa,\rho\right\}+\epsilon)^2\omega_d}{2(qa_1(\max\left\{\kappa,\rho\right\}+\epsilon)+a_2(\max\left\{\kappa,\rho\right\}+\epsilon)^q)\|K\|_{L^1({\mathbb{S}}^d)}},\frac{sq(1-s-d)\omega_d}{
2\|K\|_{L^1({\mathbb{S}}^d)}\rho^{q-2}}\right[,
$$
 where
$$
\kappa:=\Big(\frac{2}{s(1-s-d)\omega_d}\Big)^{1/2},
$$
\noindent the problem
$$-\Delta u =
\lambda|x|^{s-2}K({x}/{|x|})g(|x|^{-s}u),\ \ x\in \RR
^{d+1}\setminus \{0\},$$
admits at least three distinct nontrivial solutions.
}\end{example}

\medskip
{\bf Acknowledgments.}
The authors express their gratitude to the anonymous referees for useful comments and remarks.\par
V.~R\u adulescu acknowledges the support through  Grant CNCSIS PCCE--8/2010
``Sisteme di\-fe\-ren\-\c{t}iale \^{\i}n analiza neliniar\u{a} \c{s}i aplica\c{t}ii".

\begin{thebibliography}{99}

\bibitem{aubinjmpa} T. Aubin, {\it \'Equations diff\'erentielles non lin\'eaires et proble\`eme de Yamabe concernant la courbure scalaire},
J.~Math. Pures Appl. {\bf 55} (1976), 269--296.

\bibitem{aubin} T. Aubin, {\it Nonlinear Analysis on Manifolds. Monge--Ampe\`ere Equations}, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol.~252. Springer-Verlag, New York, 1982.

\bibitem{A-konyv} T. Aubin, {\it Some Nonlinear Problems in Riemannian Geometry},
Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998.

\bibitem{berger} M. Berger, {\it La g\'eometrie m\'etrique des vari\'et\'es Riemanniennes}, in ``Elie Cartan et les Math\'ematiques d'Aujourd'Hui", Ast\'erisque, Soci\'et\'e Math\'ematique de France 1985, pp. 9--66.

\bibitem{BV-V-Invent} M.F. Bidaut-V\'eron and L.  V\'eron, {\it Nonlinear
elliptic equations on compact Riemannian manifolds and asymptotics
of Emden equations,} {Invent. Math.}  {\bf 106} (1991), 489--539.

\bibitem{Be} W. Beckner, {\it
Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality},
Ann. of Math. \textbf{138} (1993), 213--242.

\bibitem{BONANNO}
G. Bonanno,
\newblock{\it Some remarks on a three critical points theorem},
\newblock{\rm Nonlinear Anal. TMA} {\bf 54} (2003), 651--665.

\bibitem{BONCAN}
G. Bonanno and P. Candito,
\newblock{\it Non-differentiable functionals and applications to elliptic problems with discontinuous nonlinearities},
\newblock{\rm J. Differential Equations} {\bf 244} (2008), 3031--3059.

\bibitem{BONMAR}
G. Bonanno and S.A. Marano,
\newblock{\it On the structure of the critical set of non-differentiable functions with a weak compactness condition},  Appl. Anal. \textbf{89} (2010), 1--10.

\bibitem{BOMO}
G. Bonanno and G. Molica Bisci,
\newblock{\it Three weak solutions for Dirichlet problems}, preprint (2010).

\bibitem{CI-BUll-Sci-math} A. Cotsiolis and D. Iliopoulos, {\it \'Equations elliptiques non
lin\'eaires \`a croissance de Sobolev sur-critique,} {Bull.
Sci. Math.} {\bf 119} (1995), 419--431.

\bibitem{CI-C.R.Acad} A. Cotsiolis and D. Iliopoulos,
{\it \'Equations elliptiques non lin\'eaires sur ${\mathcal S}\sp n$. Le
probl\`eme de Nirenberg,} {C. R. Acad. Sci. Paris, S\'er. I
Math.} {\bf 313} (1991), 607--609.

\bibitem{Hebey} E. Hebey, {\it Nonlinear Analysis on Manifolds: Sobolev
Spaces and Inequalities}, Courant Lecture Notes in Mathematics,
New York, 1999.

\bibitem{kazwar} J.L. Kazdan and F.W. Warner, {\it Scalar curvature and conformal deformation of Riemannian structure}, J. Differential Geometry {\bf 10} (1975), 113--134.

\bibitem{K1} A. Krist\'{a}ly, {\it Asymptotically critical problems on higher-dimensional spheres}, Discrete Contin. Dyn. Syst. \textbf{23} (2009), 919--935.

\bibitem{KM} A. Krist\'{a}ly and W. Marzantowicz, {\it Multiplicity of symmetrically distinct sequences of solutions for a quasilinear problem in ${\mathbb{R}}^N$}, Nonlinear Diff. Eqns. Appl. (NoDEA) \textbf{15} (2008), 209--226.

\bibitem{KR} A. Krist\'{a}ly and V. R\u{a}dulescu, {\it Sublinear eigenvalue problems on compact Riemannian manifolds with applications in Emden-Fowler equations}, Studia Math. 191 (2009), 237--246.

\bibitem{KRV} A. Krist\'{a}ly, V. R\u{a}dulescu and Cs. Varga, {\it Variational Principles in Mathematical Physics, Geometry, and Economics:
Qualitative Analysis of Nonlinear Equations and Unilateral Problems}, Encyclopedia of Mathematics and its Applications, No.~136, Cambridge University Press, Cambridge, 2010.

\bibitem{leepar} J.M. Lee and T.H. Parker, {\it The Yamabe problem}, Bull. Amer. Math. Soc. {\bf 17} (1987), 37--91.

\bibitem{nire} L. Nirenberg, {\it On elliptic partial differential equations}, Ann. Scuola Norm. Sup. Pisa {\bf 13} (1959), 115--162.

\bibitem{schoen} R.M. Schoen, {\it Conformal deformation of a Riemannian metric to constant scalar curvature}, J.~Differential Geometry {\bf 20} (1984), 479--495.

\bibitem{VV-C.R.Paris} J.L. V\'azquez and L. V\'eron, {\it Solutions positives
d'\'equations elliptiques semi-lin\'eaires sur des vari\'et\'es
riemanniennes compactes}, {C. R. Acad. Sci. Paris, S\'er. I
Math.} {\bf 312} (1991),  811--815.

\bibitem{yamabe} H. Yamabe, {\it On a deformation of Riemannian structures on compact manifolds}, Osaka Journal of Mathematics {\bf 12} (1960), 21--37.

\end{thebibliography}
\end{document} 