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\begin{center} \Large \bf
Sublinear eigenvalue problems on compact Riemannian manifolds with
applications in Emden-Fowler equations
\end{center}


%\titlerunning{Sublinear eigenvalue problems on compact Riemannian manifolds}

\begin{center}  {\sc
Alexandru Krist\'aly}\footnote{A. Krist\'aly is supported by
        CNCSIS grant PN II, IDEI 527.} and {\sc Vicen\c tiu
        R\u adulescu}\footnote{V. R\u{a}dulescu is partially supported by
        Grant 2-CEx06-11-18/2006.}
\end{center}

%\authorrunning{Alexandru Krist\'aly}
%\institute{A. Krist\'aly\at Babe\c s-Bolyai University, Department
%of Economics, 400591\\ Cluj-Napoca, Romania,
 %       E-mail: alexandrukristaly@yahoo.com,  and
%\and
%    --------------   \at Central European University, Department of
%Mathematics, 1051\\ Budapest, Hungary
%        }

%\date{Received:  / Accepted:  /\\
%Published online: } \maketitle



\begin{abstract}
Let $(M,g)$ be a compact Riemannian manifold without boundary,
$\dim M\geq 3,$ and a continuous function $f:\RR \to \RR$ which is
{sublinear} at infinity. By various variational approaches,
existence of multiple solutions of the eigenvalue problem
$$-\Delta_{g} \omega+\alpha (\sigma) \omega= \tilde
K(\lambda,\sigma)f(\omega),\ \sigma\in M,\ \omega\in H_1^2(M)$$
are established for certain eigenvalues $\lambda>0$, depending on
 further properties of $f$ and on explicit forms of the function %s $\alpha$ and
  $\tilde K.$  Here, $\Delta_{g}$ stands for the
Laplace-Beltrami operator on $(M,g),$ and  $\alpha,$ $\tilde K$
are smooth positive functions. These multiplicity results are then
applied to solve Emden-Fowler equations which involve sublinear
terms at infinity.
\end{abstract}

\section{Introduction and statement of main results}
Let us consider the following parametrized Emden-Fowler (or,
Lane-Emden) equation $$-\triangle u =
\lambda|x|^{s-2}K({x}/{|x|})f(|x|^{-s}u),\ \ x\in \RR
^{d+1}\setminus \{0\},\eqno(EF)_\lambda$$ where $f:\RR \to \RR $
is a continuous function, $K$ is smooth on the  $d$-dimensional
unit sphere $S^d,$ $d\geq 3,$ $s\in \RR ,$ and $\lambda>0$ is a
parameter. The equation $(EF)_\lambda$ has been extensively
 studied in the {\it pure superlinear} case, i.e., 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 of
solutions for $(EF)_\lambda$, applying either minimization or
minimax methods. Note that in the pure superlinear case the
presence of the parameter $\lambda>0$ is not relevant due to the
rescalling technique.  One of the purposes of the present paper is
to guarantee {\it multiple} solutions of $(EF)_\lambda$ for
certain $\lambda>0$ when the nonlinearity $f:\RR \to \RR$ is

 a)  {\it not} necessarily of {\it pure power} type, and

b)  {\it sublinear at infinity,} see (\ref{sublinear-cond}) below.

\noindent The solutions of $(EF)_\lambda$ are being sought in the
particular form
\begin{equation}\label{transzformacio}
u(x)=u(|x|,x/|x|)=u(r,\sigma)=r^s\omega(\sigma).
\end{equation}
Here, $(r,\sigma)\in (0,\infty)\times S^d$ are the spherical
coordinates in $\RR ^{d+1}\setminus \{0\}$. This type of
transformation is also used  by Bidaut-V\'eron and V\'eron
\cite{BV-V-Invent}, where the asymptotics of a special form of
$(EF)_\lambda$ has been studied. Throughout
(\ref{transzformacio}), the equation $(EF)_\lambda$ reduces to
$$-\Delta_{g_0} \omega+ s(1-s-d)\omega= \lambda
K(\sigma)f(\omega),\ \sigma\in S^d,
\eqno{(\ref{transzformacio})_\lambda}$$
 where $\Delta_{g_0}$ denotes the Laplace-Beltrami operator on
 $(S^d,g_0)$ and $g_0$ is the canonical metric induced from $\RR ^{d+1}.$
%$\triangle u=r^{s-2}(s(s+d-1)\omega+\triangle_{\mathbb S^d}
%\omega)$

Note in particular that when $s=-d/2$ or $s=-d/2+1,$ and
$f(t)=|t|^{\frac{4}{d-2}}t,$ %$d\geq 3,$
the existence of a smooth
solution $\omega>0$ of $(\ref{transzformacio})_\lambda$ %represents
can be viewed as an affirmative answer to the famous Yamabe
problem on $S^d$ (see also the Nirenberg problem); for these
topics we refer the reader to Aubin \cite{A-konyv},
Cotsiolis-Ilio\-po\-u\-los \cite{CI-C.R.Acad}, Hebey\cite{Hebey},
and references therein. In these cases the right hand side of
$(\ref{transzformacio})_\lambda$ involves the Sobolev critical
exponent.

As we pointed out before, our aim is to study $(EF)_\lambda$
(specially, problem $(\ref{transzformacio})_\lambda$) in the {\it
sublinear} case. Since $1-d<s<0$ implies the coercivity of the
operator $\omega\mapsto -\Delta_{g_0}\omega+s(1-s-d)\omega$, the
form of $(\ref{transzformacio})_\lambda$ motivates the study of
the following general {\it eigenvalue  problem,}  denoted  by
$(P)_\lambda,$ which constitutes the main objective of our paper:

{\it Find $\lambda\in (0,\infty)$ and $\omega\in H_1^2(M)$ such
that} $$
  -\Delta_{g} \omega+ \alpha(\sigma)\omega=
\tilde K(\lambda,\sigma)f(\omega),\ \sigma\in M,
\eqno{(\tilde{\ref{transzformacio}})_\lambda}$$ where we assume
\begin{enumerate}
{\it \item[$(A_1)$] $(M,g)$ is a smooth compact $d$-dimensional
Riemannian manifold without boundary, $d\geq 3;$
\item[$(A_2)$] $\alpha\in C^\infty(M)$ and $\tilde K\in C^\infty((0,\infty)\times M)$ are positive %%@
functions;
\item[$(f_1)$] $f:\RR \to\RR $ is locally H\"older continuous %locally Lipschitz
and sublinear at infinity, i.e.,
\begin{equation}\label{sublinear-cond}\lim_{|t|\to
\infty}\frac{f(t)}{t}=0.
\end{equation}}
\end{enumerate}
\noindent A typical case when (\ref{sublinear-cond}) holds is
\begin{enumerate}
\item[$(f_1^{q,c})$] {\it There exist $q\in (0,1)$ and $c>0$ such that $|f(t)|\leq
c|t|^{q}$ for every $t\in \RR .$}
\end{enumerate}


 We  simply say that $\omega_\lambda\in H_1^2(M)$ is a
solution of $(P)_\lambda$ if $\omega=\omega_\lambda$ satisfies
$(\tilde{\ref{transzformacio}})_\lambda.$ Above, $\Delta_{g}$ is
the Laplace-Beltrami operator on $(M,g);$ its expression in local
coordinates is
$\Delta_{g}\omega=g^{ij}(\partial_{ij}\omega-\Gamma_{ij}^k\partial_k
\omega).$ $H_1^2(M)$ is the usual Sobolev space on $M,$ endowed
with its natural norm $\|\cdot\|_{H_1^2}$ (see \cite{Hebey} or
Section \ref{sect-aux} for details).

%Due to the sublinearity of $f$ at infinity, see
%(\ref{sublinear-cond}),
The presence of the parameter $\lambda>0$ in $(P)_\lambda$ is
indispensable. Indeed, if we consider a sublinear function at
infinity which is, in addition, uniformly Lipschitz (with
Lipschitz constant $L>0$), and $\tilde K(\lambda,\sigma)=\lambda
K(\sigma),$ with $K\in C^\infty(M)$ positive, one can prove that
for $0<\lambda<\frac{1}{L}\frac{\min_M\alpha}{\max_MK
}:=\lambda_L$ we have only the $\omega=\omega_\lambda=0$ solution
of $(P)_\lambda,$ as the standard contraction principle on the
Hilbert space $H_1^2(M)$ shows. For a concrete example, let us
consider the
 function $f(t)=\ln(1+t^2)$ and assume that $\frac{K(\sigma)}{\alpha(\sigma)}={\rm
const.}=\mu_0\in (0,\infty).$ Then, for every
$0<\lambda<\frac{\min_M\alpha}{\mu_0\max_M \alpha},$ problem
$(P)_\lambda$ has only the trivial solution; however, when
 $\lambda >\frac{5}{4\mu_0},$ problem
$(P)_\lambda$ has {\it three} distinct {constant} solutions which
are precisely  the fixed points of the function $t\mapsto
\lambda\mu_0\ln(1+t^2).$ Note that one of the solutions is the
trivial one.

%\begin{remark}\label{fixed-point-solution}
In the generic case, for a fixed $\lambda>0$, the function
$\omega_\lambda(\sigma)=c\in \RR $ is a solution of $(P)_\lambda$
if and only if $\alpha(\sigma)c=\tilde K(\lambda,\sigma)f(c)$ for
a.e.  $\sigma \in M.$ In particular, when
$\omega_\lambda(\sigma)=c\neq 0,$ the function $\sigma\mapsto
\frac{\tilde K(\lambda,\sigma)}{\alpha(\sigma)}$ is constant; let
us denote this value by $\mu_\lambda>0.$ Thus, nonzero constant
solutions of $(P)_\lambda$ appear as fixed points of the function
$t\mapsto \mu_\lambda f(t).$
%\end{remark}

In order to obtain multiple solutions of $(P)_\lambda$  not only
in the case when $\sigma\mapsto \frac{\tilde
K(\lambda,\sigma)}{\alpha(\sigma)}$ is constant for certain
$\lambda>0,$ we will use variational arguments; weak solutions of
$(P)_\lambda$ will be found as critical points of the energy
functional associated with $(P)_\lambda,$ see Section
\ref{sect-aux}. Due to $(f_1),$ these elements are actually
classical solutions of $(P)_\lambda.$

Our first result concerns the case when $f:\RR \to \RR $ fulfills
the hypothesis
\begin{enumerate}
\item[$(f_2)$] {\it $\lim_{t\to
0}\frac{f(t)}{t}=0.$}
\end{enumerate}
Before to state this result, let us note that the usual norm on
the space $L^p(M)$ will be denoted by $\|\cdot\|_p,$ $p\in
[1,\infty].$

\begin{theorem}\label{theorem-11}
Assume that $f:\RR \to \RR $ fulfills $(f_1),$ $(f_2)$ and
$\sup_{t\in \RR } F(t)>0,$ where $F(t)=\int_0^t f(\tau)d\tau.$
Assume also that $(A_1)$ and $(A_2)$ are verified with $\tilde
K(\lambda,\sigma)=\lambda K(\sigma),$ and $K\in C^\infty(M)$ is
positive. Then, there exists $\tilde \lambda>0$ such that
 for every $\lambda>\tilde \lambda$ problem $(P)_\lambda$ has at least
two distinct, nontrivial solutions.
%$\omega_\lambda^1,\omega_\lambda^2\in H_1^2(M),$ where
%$\omega_\lambda^1$ is the global minimum of the energy functional
%associated with $(P)_\lambda$, see $(\ref{energy-funct});$
% if $(f_1^{q,c})$ holds then
%$\|\omega_\lambda^1\|_{H_1^2}=o(\lambda^\frac{1}{1-r})$ for every
%$r\in (q,1),$ but $\|\omega_\lambda^1\|_{H_1^2}\neq
%%{O}(\lambda^\frac{1}{1-\mu})$ for any $\mu\in (1,(d+2)/(d-2))$ as
%$\lambda\to\infty.$
\end{theorem}

%:=\frac{1}{2}\frac{\|\alpha\|_{1}}{\|K\|_{1}}\left(\max_{t\neq 0}\frac{F(t)}{t^2}\right)^{-1}


\begin{remark}\rm\label{rem-elso}
As we mentioned above, when the nonlinearity $f$ is a uniformly
Lipschitz function (with Lipschitz constant $L>0$), we have extra
information on the eigenvalues:

(a) problem $(P)_\lambda$ has only the trivial solution whenever
$\lambda\in (0,\lambda_L)$;

(b) problem $(P)_\lambda$ has at least two nontrivial solutions
whenever $\lambda>\tilde \lambda.$

\noindent Moreover, the proof of Theorem \ref{theorem-11} shows
that the number $\tilde \lambda$ is less or equal than the value
 $\lambda^*
:=\frac{1}{2}\frac{\|\alpha\|_{1}}{\|K\|_{1}}\left(\max_{t\neq
0}\frac{F(t)}{t^2}\right)^{-1}$. Consequently, Theorem
\ref{theorem-11} is valid for any $\lambda>\lambda^*$ which can be
easily computed.
\end{remark}

 A direct consequence of Theorem \ref{theorem-11} applied
for $(\ref{transzformacio})_\lambda$ is the following

\begin{theorem}\label{theorem-12}
Assume that $1-d<s<0.$ Let $f:\RR \to \RR $ be a function as in
Theorem $\ref{theorem-11}$ and $K\in C^\infty(S^d)$ positive.
Then, there exists $\tilde \lambda>0$ such that  for every
$\lambda>\tilde \lambda$ problem $(EF)_\lambda$ has at least two
distinct, nontrivial solutions.
\end{theorem}





In order to obtain a new kind of multiplicity result concerning
$(P)_\lambda$ (specially, $(\ref{transzformacio})_\lambda$ and
$(EF)_\lambda$), we require:
\begin{enumerate}
\item[$(f_3)$] {\it There exists $\mu_0\in (0,\infty)$ such that the
global minima of the function $t\mapsto \tilde F_{\mu_0}
(t):=\frac{1}{2}t^2-\mu_0 F(t)$ has at least $m\geq 2$ connected
components.}
\end{enumerate}

\noindent Note that $(f_3)$ implies that the function $t\mapsto
\tilde F_{\mu_0} (t)$ has at least $m-1$ local maxima. Thus, the
function $t\mapsto \mu_0f (t)$ has at least $2m-1$ fixed points.
In particular, if for some $\lambda>0$ one has $\frac{\tilde
K(\lambda,\sigma)}{\alpha(\sigma)}=\mu_0$ for every $\sigma\in M,$
then problem $(P)_{\lambda}$ has at least $2m-1\geq 3$ constant
solutions. On the other hand, the following general result can be
shown.

\begin{theorem}\label{theorem-15}
Let $f:\RR \to \RR $ be a function which fulfills $(f_1)$ and
$(f_3).$  Assume that $(A_1)$ and $(A_2)$ are verified with
$\tilde K(\lambda,\sigma)= \lambda K(\sigma)+\mu_0\alpha(\sigma),$
and $K\in C^\infty(M)$ is positive. Then
\begin{enumerate}
\item[a)]  for every $\eta>\max\{0,\|\alpha\|_{1}\min_{t\in
\RR  }\tilde F_{\mu_0} (t)\}$ there exists a number $\tilde
\lambda_\eta>0$ such that for every $\lambda\in (0,\tilde
\lambda_\eta)$ problem $(P)_\lambda$ has at least $m+1$ solutions
$\omega_\lambda^{1,\eta},\dots,\omega_\lambda^{m+1,\eta}\in
H_1^2(M);$
\item[b)] if  $(f_1^{q,c})$ holds  then for each
$\lambda\in (0,\tilde \lambda_\eta)$ there is a set
$I_\lambda\subset \{1,\dots,m+1\}$ with {\rm card}$(I_\lambda)=m$
such that  $$
\|\omega_\lambda^{i,\eta}\|_{H_1^2}<\frac{t_{\eta,q,c}}{\min\{1,{\min_{M}\alpha^{1/2}}\}},\
\ \ \ \ i\in I_\lambda,$$ where $t_{\eta,q,c}>0$ is the greatest
solution of the equation $$\frac{1}{2}t^2-\frac{\mu_0
c\|\alpha\|_{1}^{(1-q)/2}}{q+1}t^{q+1}-\eta=0,\ \ \ t>0.$$
\end{enumerate}
\end{theorem}

\noindent A consequence of Theorem \ref{theorem-15} in the context
of $(EF)_\lambda$ reads as follows.

\begin{theorem}\label{theorem-16}
Assume that  $1-d<s<0.$ Let $f:\RR \to \RR $ be a function as in
Theorem $\ref{theorem-15}$ and $K\in C^\infty(S^d)$ be a positive
function.  Then, there exists $\lambda_0>0$ such that for every
$\lambda\in (0,\lambda_0)$ problem $$-\triangle u =
|x|^{s-2}[\lambda K(x/|x|)+\mu_0 s(1-s-d)] f(|x|^{-s}u),\ \ x\in
\RR^{d+1}\setminus \{0\}$$
 has at least $m+1$ solutions.
\end{theorem}

\noindent The proof of Theorem \ref{theorem-11} is based on the
paper of Pucci-Serrin \cite{PS}, while that of Theorem
\ref{theorem-15} relies on a recent abstract critical point
theorem of Ricceri, see \cite{Ri-3} and \cite{Ric-PAMS-Neumann}.\\

\noindent {\bf Examples.}

a) Let $f(t)=\ln (1+t^2).$ One can apply Theorems
\ref{theorem-11}--\ref{theorem-12}.


b)  Let $f:\RR \to \RR $ be defined by $f(t)=\min\{t_+-\sin(\pi
t_+),2(m-1)\}$ where $m\in \NN\setminus\{1\}$ is fixed and we use
the notation $t_+=\max\{t,0\}.$ Clearly, $(f_1)$ is verified,
while for $\mu_0=1,$ the assumption $(f_3)$ is also fulfilled.
Indeed, the function $t\mapsto \tilde F_1(t)$ has precisely $m$
global minima; they are $0,2,\dots, 2(m-1).$ Moreover, $\min_{t\in
\RR }\tilde F_{1} (t)=0.$ Therefore, one can apply Theorems
\ref{theorem-15} and \ref{theorem-16}.\\

In the next section we  recall some basic facts on the Sobolev
spaces defined on compact Riemannian manifolds. In Section
\ref{elsotetel-biz} we will prove Theorems \ref{theorem-11} and
\ref{theorem-12} while in the last section we are dealing with the
proofs of Theorems \ref{theorem-15} and \ref{theorem-16}.



\section{Preliminaries}\label{sect-aux}

 Let $(M,g)$ be a smooth compact $d$-dimensional Riemannian manifold without
boundary, $d\geq 3,$ and let $\alpha\in C^\infty(M)$ be a positive
function. For every $\omega\in C^\infty (M),$ set
$$\|\omega\|_{H_\alpha^2}^2=\int_M \langle \nabla \omega, \nabla
\omega \rangle d\sigma_g+\int_M\alpha(\sigma)\langle \omega,\omega
\rangle d\sigma_g,$$ where $\langle \cdot,\cdot \rangle$ is the
inner product on covariant tensor fields associated to $g,$
$\nabla \omega$ is the covariant derivative of $\omega,$ and
$d\sigma_g$ is the Riemannian measure. The Sobolev space
$H_\alpha^2(M)$ is defined as the completion of $C^\infty(M)$ with
respect to the norm $\|\cdot\|_{H_\alpha^2}.$ Clearly,
$H_\alpha^2(M)$ is a Hilbert space endowed with the inner product
$$\langle \omega_1,\omega_2\rangle_{H_\alpha^2}=\int_M \langle
\nabla \omega_1,\nabla \omega_2 \rangle
d\sigma_g+\int_M\alpha(\sigma)\langle \omega_1,\omega_2 \rangle
d\sigma_g, \ \ \ \ \ \omega_1,\omega_2\in H_\alpha ^2(M).$$ Since
$\alpha$ is positive, the norm $\|\cdot \|_{H_\alpha^2}$ is
equivalent with the standard norm $\|\cdot \|_{H_1^2};$ actually,
the latter norm is nothing but $\|\cdot \|_{H_\alpha^2}$ with
$\alpha=1.$ Moreover, we have
\begin{equation}\label{equivalance}
\small
  \min\{1,{\min_{M}\alpha^{1/2}}\}\|\omega\|_{H_1^2}\leq
\|\omega\|_{H_\alpha^2} \leq
\max\{1,{\|\alpha\|_\infty^{1/2}}\}\|\omega\|_{H_1^2},\ \ \
\omega\in H_\alpha^2(M).
\end{equation}
 Note that $H_\alpha ^2(M)$ is compactly
embedded into $L^p(M)$ for every $p\in [1,2d/(d-2));$ the Sobolev
embedding constant will be denoted by $S_p>0.$

 Let $\lambda>0.$ The energy functional $\mathcal E_\lambda:H_1^2(M)\to
\RR $ associated with problem $(P)_\lambda$ is
\begin{equation}\label{energy-funct}
  \mathcal
E_\lambda(\omega)=\frac{1}{2}\|\omega\|_{H_\alpha^2}^2-\int_M
\tilde K(\lambda,\sigma)F(\omega(\sigma))d\sigma_g,
\end{equation}
where $F(t)=\int_0^t f(\tau)d\tau.$ Due to our initial assumptions
$(A_1),$ $(A_2)$ and $(f_1),$ the functional $\mathcal E_\lambda$
is well-defined, it belongs to $C^1(H_1^2(M),\RR ),$ and its
critical points are precisely the weak (so, classical) solutions
of problem $(P)_\lambda.$ By $(f_1)$, 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 $\omega\in
 H_1^2(M),$ we have $$\mathcal
E_\lambda(\omega)\geq\frac{1}{2}(1-\varepsilon\|\tilde
K(\lambda,\cdot)\|_\infty
S_2^2)\|\omega\|_{H_\alpha^2}^2-c(\varepsilon)\|\tilde
K(\lambda,\cdot)\|_\infty S_1\|\omega\|_{H_\alpha^2}. $$
Therefore, the  functional $\mathcal E_\lambda$
 is coercive and bounded from below on $H_1^2(M).$ Moreover, it satisfies the
 standard Palais-Smale condition, see Zeidler \cite[Example
 38.25]{Zei}.



\section{Proof of Theorems \ref{theorem-11} and \ref{theorem-12}}\label{elsotetel-biz}


In this section we assume the hypotheses of Theorem
 \ref{theorem-11} are fulfilled. We introduce the functionals $\mathcal N,\mathcal F:H_1^2(M)\to %%@
\RR $ defined by
\begin{equation}\label{N-F-fuggvenyek}
  \mathcal N(\omega)=\frac{1}{2}\|\omega\|_{H_\alpha^2}^2\ \ \ {\rm and} \ \ \
  \mathcal F(\omega)=\int_M K(\sigma)F(\omega(\sigma))d\sigma_g,\ \ \omega\in H_1^2(M).
\end{equation}


\begin{proposition}\label{prop-41}
$\lim_{\rho\to 0^+}\sup\left\{\frac{\mathcal F(\omega)}{\mathcal
N(\omega)}:0<\mathcal N(\omega)<\rho\right\}=0.$
\end{proposition}
{\it Proof.} Due to $(f_2),$ for an arbitrarily small
$\varepsilon>0$ there exists $\delta(\varepsilon)>0$ such that
$|f(t)|<{\varepsilon}({2}\|K\|_\infty S_2^{2})^{-1}|t|$ for every
$|t|<\delta(\varepsilon).$ On account of $(f_1)$, one may fix
$1<\nu<\frac{d+2}{d-2}$ and $c(\varepsilon)>0$ such that
$|f(t)|<c({\varepsilon})|t|^\nu$ for every $|t|\geq
\delta(\varepsilon).$ Combining these two facts, after an
integration, we obtain
%\begin{equation}\label{F-becs-zero}
$$|F(t)|\leq {\varepsilon}({4}\|K\|_\infty
S_2^{2})^{-1}t^2+c(\varepsilon)(\nu+1)^{-1}|t|^{\nu+1}\ \ \ {\rm
for\ every}\ t\in \RR .$$
%\end{equation}
Fix a $\rho>0$ and any element $\omega\in H_\alpha^2(M)$ complying
with $\mathcal N(\omega)<\rho.$ Due to the above estimation, we
have
\begin{eqnarray*}
\mathcal F(\omega)&\leq&
\frac{\varepsilon}{4}\|\omega\|_{H_\alpha^2}^2+\frac{c(\varepsilon)}{\nu+1}\|K\|_\infty
S_{\nu+1}^{\nu+1}\|\omega\|_{H_\alpha^2}^{\nu+1}\\ &<&
\frac{\varepsilon}{4}\|\omega\|_{H_\alpha^2}^2+\frac{c(\varepsilon)}{\nu+1}\|K\|_\infty
S_{\nu+1}^{\nu+1} (2\rho)^\frac{\nu-1}{2}\|\omega\|_{H_\alpha^2}^2
\\ &=&
 \left(\frac{\varepsilon}{4}+c'(\varepsilon) \rho^\frac{\nu-1}{2}\right)
\|\omega\|_{H_\alpha^2}^2.
\end{eqnarray*} Thus there exists $\rho(\varepsilon)>0$ such that
for every $0<\rho<\rho(\varepsilon)$, we have  $$0\leq
\sup\left\{\frac{\mathcal F(\omega)}{\mathcal
N(\omega)}:0<\mathcal N(\omega)<\rho\right\}\leq
\frac{\varepsilon}{2} +
2c'(\varepsilon)\rho^\frac{\nu-1}{2}<\varepsilon, $$ which
completes the proof.\\


{\it Proof of Theorem \ref{theorem-11}.} Let us define $$\tilde
\lambda=\inf\left\{\frac{\mathcal N(\omega)}{\mathcal
F(\omega)}:\mathcal F(\omega)>0\right\}.$$ Fix $\lambda>\tilde
\lambda$. On one hand, there exists $\omega_\lambda^0\in H_1^2(M)$
such that $\mathcal F(\omega_\lambda^0)>0$ and
$\lambda>\frac{\mathcal N(\omega_\lambda^0)}{\mathcal
F(\omega_\lambda^0)}.$ On account of (\ref{energy-funct}) and
(\ref{N-F-fuggvenyek}), we have
\begin{equation}\label{energia-kisebb-mint-nulla}
\mathcal E_\lambda(\omega_\lambda^0)=\mathcal
N(\omega_\lambda^0)-\lambda\mathcal F(\omega_\lambda^0)<0.
\end{equation}
On the other hand, due to Proposition \ref{prop-41}, there exists
$\delta>0$ such that for all $\rho\in (0,\delta)$ one has
$$\sup\left\{\frac{\mathcal F(\omega)}{\mathcal
N(\omega)}:0<\mathcal N(\omega)<\rho\right\}<\frac{1}{\lambda}.$$
In other words, for every $\omega\in H_1^2(M)\setminus \{0\}$ with
$\mathcal N(\omega)<\rho$ we have $\frac{\mathcal
F(\omega)}{\mathcal N(\omega)}<\frac{1}{\lambda}$ and hence
$$\mathcal E_\lambda(\omega)=\mathcal N(\omega)-\lambda\mathcal
F(\omega)>0=\mathcal E_\lambda(0).$$ Consequently, $0\in H_1^2(M)$
is a local minimum for $\mathcal E_\lambda$, but not a global one,
taking into account relation (\ref{energia-kisebb-mint-nulla}).
Since the  functional $\mathcal E_\lambda$  is bounded from below
on $H_1^2(M)$ and  it satisfies the  standard Palais-Smale
condition, the global minimum of $\mathcal E_\lambda$ is achieved.
Applying \cite[Theorems 1 and 4]{PS}, we obtain a third critical
point of $\mathcal E_\lambda$ which is not $0$. This concludes our
proof.\\

\noindent {\it Proof of Theorem \ref{theorem-12}.} Let us choose
$(M,g)=(S^d,g_0),$ and $\alpha(\sigma)=s(1-s-d)$ for every
$\sigma\in S^d$ in Theorem \ref{theorem-11}.   Thus, for every
$\lambda>\tilde \lambda$,  problem
$(\ref{transzformacio})_\lambda$ has at least two distinct,
nontrivial solutions $\omega_\lambda^1,\omega_\lambda^2\in
H_1^2(S^d).$ On account of (\ref{transzformacio}), the elements
$u_\lambda^i(x)=|x|^s\omega_\lambda^i(x/|x|),$ $i\in \{1,2\},$ are
solutions of $(EF)_\lambda.$

\begin{remark} \rm\label{O(d+1)-problema}
For every $k\in \{ 1,\dots,d-2\},$ let $G_k=O(k+1)\times
O(d-k)\subset O(d+1).$
 The action of the group $G_k$ on
 $H_1^2(S^d)$ is defined as $g\omega (\sigma)=\omega(g^{-1}\sigma)$ for
 every $g\in G_k,$ $\omega\in H_1^2(S^d),$ $\sigma\in
 S^d.$ Assume that $K$ is constant in
 $(\ref{transzformacio})_\lambda.$ One can prove that the energy
 functional $\mathcal E_\lambda:H_1^2(S^d)\to \RR $ is
 $G_k$-invariant for every $k\in \{ 1,\dots,d-2\},$ i.e.,
 $\mathcal E_\lambda(g\omega)= \mathcal E_\lambda(\omega),$ for
 every $g\in G_k$ and $\omega\in H_1^2(S^d).$ Now, let
%\begin{equation}\label{H_G_k-definicio}
$$   H_{G_k}(S^d)=\{\omega\in H_1^2(S^d):g\omega =\omega\ {\rm
for\ every}\ g\in G_k\}$$ be the fixed point space of $H_1^2(S^d)$
under the action of $G_k$. Let $\mathcal E_\lambda^{G_k}$ be the
restriction of the functional $\mathcal E_\lambda$ to the space
$H_{G_k}(S^d).$ Now, we follow the same arguments as in Theorems
\ref{theorem-11} with $\mathcal E_\lambda^{G_k}$ and
$H_{G_k}(S^d)$ instead of $\mathcal E_\lambda$ and $H_1^2(S^d),$
respectively. Therefore, for every $\lambda>\tilde \lambda$ and
$k\in \{ 1,\dots,d-2\}$ we can guarantee the existence of at least
two
 different, nontrivial critical points of $\mathcal
E_\lambda^{G_k}$ belonging to $H_{G_k}(S^d).$ On the other hand,
the principle of symmetric criticality of Palais implies that
every critical point of $\mathcal E_\lambda^{G_k}$ will be also a
critical point of $\mathcal E_\lambda,$  thus a solution of
$(\ref{transzformacio})_\lambda.$
 If such a solution is {not} constant, say $\omega_{\lambda,k}\in H_{G_k}(S^d),$ it cannot belong
 to $H_{G_l}(S^d)$ whenever $l\neq k.$
 Indeed, when $k\neq l,$ the group generated
topologically by $G_k$ and $G_l$
 acts transitively on $S^d;$ namely, the orbit of every element
 from $S^d$ by the group generated by $G_k$ and $G_l$ is the whole
 sphere $S^d.$  Consequently,
  $H_{G_k}(S^d)\cap H_{G_l}(S^d)$ contains only the a.e. constant  functions defined on
 $S^d.$  Therefore, a non-constant solution of $(\ref{transzformacio})_\lambda$
 which belongs to $H_{G_k}(S^d)$ will not appear in $H_{G_l}(S^d),$ $l\neq k.$
 In this way, the number of solutions of $(\ref{transzformacio})_\lambda$ and $(EF)_\lambda$
 can increase.
 \end{remark}



\section{Proof of Theorems \ref{theorem-15} and \ref{theorem-16}}\label{haromtetel-biz}

We assume the hypotheses of Theorem
 \ref{theorem-15} are fulfilled. Using the notation from the previous section, see
 (\ref{N-F-fuggvenyek}), we define the functional $\mathcal
 N_{\mu_0}:H_1^2(M)\to \RR $ by
 $$\mathcal  N_{\mu_0}(\omega)=\mathcal
 N(\omega)-{\mu_0}\int_M \alpha(\sigma)F(\omega(\sigma))d\sigma_g, \ \ \omega\in H_1^2(M).$$

\begin{proposition}\label{lok-minim} The set of all global minima of the functional $\mathcal %%@
N_{\mu_0}$ has at least $m$ connected components in the weak
topology on $H_1^2(M).$
\end{proposition}

\noindent {\it Proof.} First, for every $\omega\in H_1^2(M)$ we
have
\begin{eqnarray*}
\mathcal N_{\mu_0}(\omega)&= &
\frac{1}{2}\|\omega\|_{H_\alpha^2}^2-{\mu_0}\int_M
\alpha(\sigma)F(\omega(\sigma))d\sigma_g\\&= &
\frac{1}{2}\int_M|\nabla\omega|^2d\sigma_g+\int_M
\alpha(\sigma)\tilde F_{\mu_0}(\omega(\sigma))d\sigma_g\\&\geq &
\|\alpha\|_1\inf_{t\in \RR }\tilde F_{\mu_0}(t).
\end{eqnarray*}
Moreover, if we consider $\omega(\sigma)=\omega_{\tilde
t}(\sigma)=\tilde t$ for a.e. $\sigma \in M,$ where $\tilde t\in
\RR $ is a minimum point of the function $t\mapsto \tilde
F_{\mu_0}(t),$ then we have equality in the previous estimation.
Thus,  $$\inf_{\omega\in H_1^2(M)}\mathcal N_{\mu_0}(\omega)=
\|\alpha\|_1\inf_{t\in \RR }\tilde F_{\mu_0}(t).$$ Moreover, if
$\omega\in H_1^2(M)$ is not a constant function, then $|\nabla
\omega|^2=g^{ij}\partial_i\omega
\partial_j\omega>0$ on a positive measured set of the manifold
$M.$ In this case, we have $$\mathcal
N_{\mu_0}(\omega)=\frac{1}{2}\int_M|\nabla\omega|^2d\sigma_g+\int_M
\alpha(\sigma)\tilde
F_{\mu_0}(\omega(\sigma))d\sigma_g>\|\alpha\|_1\inf_{t\in \RR}
\tilde F_{\mu_0}(t).$$ Consequently, between the sets $${\rm
Min}({\mathcal N_{\mu_0}})=\{\omega\in H_1^2(M):\mathcal
N_{\mu_0}(\omega)=\inf_{\omega\in H_1^2(M)}\mathcal
N_{\mu_0}(\omega)\}$$ and $${\rm Min}(\tilde F_{\mu_0})=\{t\in \RR
:\tilde F_{\mu_0}(t)=\inf_{t\in \RR }\tilde F_{\mu_0}(t)\}$$ there
is a one-to-one correspondence. Indeed, let $\theta$ be the
function that associates to every number $t\in \RR $ the
equivalence class of those functions which are a.e. equal to $t$
in the whole manifold $M.$ Then $\theta:{\rm Min}(\tilde
F_{\mu_0})\to {\rm Min}({\mathcal N_{\mu_0}})$ is actually a
homeomorphism between ${\rm Min}(\tilde F_{\mu_0})$ and ${\rm
Min}({\mathcal N_{\mu_0}})$, where the set ${\rm Min}({\mathcal
N_{\mu_0}})$ is considered with the relativization of the weak
topology on $H_1^2(M).$ On account of the hypothesis $(f_3),$ the
set ${\rm Min}(\tilde F_{\mu_0})$ contains at least $m\geq 2$
connected components. Therefore, the same is true for the set
${\rm Min}({\mathcal N_{\mu_0}})$, which completes the proof.\\

\noindent Since $\tilde K(\lambda,\sigma)= \lambda
K(\sigma)+\mu_0\alpha(\sigma),$ the energy functional associated
to $(P)_\lambda$ has the form $\mathcal E_\lambda=\mathcal
N_{\mu_0}-\lambda \mathcal F,$ where $\mathcal F$ comes from
(\ref{N-F-fuggvenyek}). In order to prove Theorem
\ref{theorem-15}, we recall a recent critical point result of
Ricceri \cite[Theorem 8]{Ri-3}.

\begin{theorem}\label{Ric-masodik} Let $H$ be a separable and reflexive real Banach space, and let
$\mathcal N,\mathcal G:H\to \RR $ be two sequentially weakly lower
semicontinuous and continuously G\^ateaux differentiable
functionals, with $\mathcal N$ coercive. Assume that the
functional $\mathcal N+\lambda\mathcal G$ satisfies the
Palais-Smale condition for every $\lambda>0$ small enough and that
the set of all global minima of $\mathcal N$ has at least $m$
connected components in the weak topology, with $m\geq 2.$

Then, for every $\eta>inf_H \mathcal N,$ there exists $\overline
\lambda>0$ such that for every $\lambda\in (0,\overline \lambda),$
the functional $\mathcal N+\lambda\mathcal G$ has at least $m+1$
critical points, $m$ of which are lying in the set $\mathcal
N^{-1}((-\infty,\eta)).$
\end{theorem}

\noindent {\it Proof of Theorem \ref{theorem-15}.} Let us choose
$H=H_1^2(M),$   $\mathcal N=\mathcal N_{\mu_0}$ and $\mathcal
G=-\mathcal F$ in Theorem \ref{Ric-masodik}. Due to Proposition
\ref{lok-minim} and to basic properties of the functions $\mathcal
N_{\mu_0},$  $\mathcal F$, all the hypotheses of Theorem
\ref{Ric-masodik} are satisfied.

Then, for every $\eta>\max\{0,\|\alpha\|_{1}\min_{t\in \RR }\tilde
F_{\mu_0} (t)\}\left(\geq \inf_{\omega\in H_1^2(M)}\mathcal
N_{\mu_0}(\omega)\}\right)$ there is a number $\tilde
\lambda_\eta>0$ such that for every $\lambda\in (0,\tilde
\lambda_\eta)$ the function $\mathcal N_{\mu_0}-\lambda \mathcal
F$ has at least $m+1$ critical points; let us denote them by
$\omega_\lambda^{1,\eta},\dots,\omega_\lambda^{m+1,\eta}\in
H_1^2(M).$ Clearly, they are solutions of problem $(P)_\lambda,$
which concludes $a).$

We know in addition that $m$ elements from
$\omega_\lambda^{1,\eta},\dots,\omega_\lambda^{m+1,\eta}$  belong
to the set $\mathcal N_{\mu_0}^{-1}((-\infty,\eta)).$ Let
$\tilde\omega$ be such an element, i.e.,
\begin{equation}\label{N-kisebb}
  \mathcal
N_{\mu_0}(\tilde\omega)=\frac{1}{2}\|\tilde\omega\|_{H_\alpha^2}^2-{\mu_0}\int_M
\alpha(\sigma)F(\tilde\omega(\sigma))d\sigma_g<\eta.
\end{equation}
 Assume that
 $(f_1^{q,c})$ holds. Then $|F(t)|\leq \frac{c}{q+1}|t|^{q+1}$
for every $t\in \RR .$ By using the H\"older inequality, one has
\begin{equation}\label{estimat-alpha}
  \int_M \alpha(\sigma)|\tilde\omega(\sigma)|^{q+1}d\sigma_g\leq
\|\alpha\|_1^{(1-q)/2}\|\tilde\omega\|_{H_\alpha^2}^{q+1}.
\end{equation}
Since $\eta>0,$ the equation
\begin{equation}\label{utolso-egy}
  \frac{1}{2}t^2-\frac{\mu_0
c\|\alpha\|_{1}^{(1-q)/2}}{q+1}|t|^{q+1}-\eta=0,
\end{equation}
always has a positive solution. On account of (\ref{N-kisebb}) and
(\ref{estimat-alpha}), the number $\|\tilde\omega\|_{H_\alpha^2}$
is less then the  greatest solution $t_{\eta,q,c}>0$ of the
equation (\ref{utolso-egy}). It remains to apply
(\ref{equivalance}), which concludes $b).$\\

\noindent {\it Proof of Theorem \ref{theorem-16}.} It follows
directly by Theorem \ref{theorem-15}.

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\noindent \small {\sc Alexandru Krist\'aly,}  University of Babe\c
s-Bolyai, Department of Economics, 400591 Cluj-Napoca, Romania.\\
{\it Email address}: {\tt alexandrukristaly@yahoo.com}\\
\\
{\sc Vicen\c tiu  R\u adulescu,}  Institute of Mathematics
``Simion Stoilow" of the Romanian Academy, 014700 Bucharest,
Romania; and\\ University of Craiova, Department of Mathematics,
200585 Craiova, Romania.\\ {\it Email addresses}: {\tt
vicentiu.radulescu@math.cnrs.fr}\\ {\it URL}: {\tt
http://www.inf.ucv.ro/\~{}radulescu}


\end{document}
