\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 123, pp. 1--17.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/123\hfil 
Existence of infinitely many  symmetric solutions]
{Existence of infinitely many symmetric solutions to perturbed elliptic equations
 with discontinuous nonlinearities in $\mathbb{R}^N$}

\author[S. Heidarkhani, F. Gharehgazlouei, A. Solimaninia \hfil EJDE-2015/123\hfilneg]
{Shapour Heidarkhani, Fariba Gharehgazlouei, Arezoo Solimaninia}

\address{Shapour Heidarkhani \newline
Department of Mathematics, Faculty of Sciences,
Razi University,  67149 Kermanshah, Iran}
\email{s.heidarkhani@razi.ac.ir}

\address{Fariba Gharehgazlouei \newline
Department of Mathematics, Faculty of Sciences,
Razi University,  67149 Kermanshah, Iran}
\email{f.gharehgazloo@yahoo.com}

\address{Arezoo Solimaninia \newline
Department of Mathematics, Faculty of Sciences,
Razi University,  67149 Kermanshah, Iran}
\email{a.solimaninia@yahoo.com}

\thanks{Submitted November 10, 2014. Published May 5, 2015.}
\subjclass[2010]{34A16, 35J20}
\keywords{Radially symmetric solutions; perturbed elliptic equation;
\hfill\break\indent 
 discontinuous nonlinearities; critical point theory; variational method}


\begin{abstract}
 In this article we study the existence of infinitely many radially
 symmetric solutions for a class of perturbed elliptic equations
 with discontinuous nonlinearities in $\mathbb{R}^N$.
 We determine open intervals of positive parameters for which the
 problem admits infinitely many symmetric solutions. Our proofs are based on
 variational methods.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks

\section{Introduction}

 We consider the perturbed elliptic problem
\begin{equation} \label{1}
 -\Delta_{p}u+| u |^{p-2}u=\lambda f(| x|, u)
 +\mu g(| x|, u),
 \quad  x\in \mathbb{R}^{N},\quad u \in W_{r}^{1,p}(\mathbb{R}^{N})
\end{equation}
where  $\Delta_{p}u:=\operatorname{div}(|\nabla u|^{p-2}\nabla u)$ is the
$p$-Laplacian operator, $\lambda >0$, $\mu\geq 0$,
$2\leq N<p<+\infty$, the functions 
$f,g:\mathbb{R}^{N}\times\mathbb{R}\to \mathbb{R}$
are continuous almost everywhere.
We recall that $f$ is  continuous almost everywhere if the set
$D_{f}=\cup_{x\in \mathbb{R}^{N}}\{ z\in \mathbb{R}: f(|x|, .)
 \text{ is discontinuous at }  z\}$ has measure zero.

 Since many free boundary problems and obstacle problems may be
reduced to partial differential equations with discontinuous
nonlinearities, as it arises in physics problems, such as
nonlinear elasticity theory, mechanics and engineering topics, the
existence of multiple solutions for Dirichlet boundary value
problems with discontinuous nonlinearities has been widely
investigated in recent years. Chang  \cite{Ch} extended the
variational methods to a class of non-differentiable functionals,
and applied directly the variational methods for
non-differentiable functionals to prove some existence theorems
for PDE with discontinuous nonlinearities. Later, Hu et al. in
\cite{HKP} obtained the existence of two solutions for an
eigenvalue Dirichlet problem involving the $p$-Laplacian with
discontinuous nonlinearities. Next, Motreanu and Panagiotopoulos
 \cite[ Chapter 3]{MotPan} studied the critical point theory
for non-smooth functionals and in this framework, very recently,
Marano and Motreanu \cite{MM} obtained an infinitely many
critical points theorem, which extends the Variational Principle
of Ricceri \cite{R} to non-smooth functionals, and applies the
result to variational-hemivariational inequalities and semilinear
elliptic eigenvalue problems with discontinuous nonlinearities. In
\cite{BM2} Bonanno and Molica Bisci presented a more precise
version of the infinitely many critical points theorem of Marano
and Motreanu, and as an application of their result, they ensured
the existence of infinitely many solutions for a two-point
boundary value problem with the Sturm-Liouville equation having
discontinuous nonlinear term. In \cite{K1} the authors employing
the same critical points theorem of Marano and Motreanu,
investigated the existence of infinitely many radially symmetric
solutions for a class of differential inclusion problems. In
\cite{ZL} the authors using a three critical points theorem for a
non-differentiable functional and a Sobolev embedding result,
established the existence of three radially symmetric solutions
for the problem \eqref{1}, in the case $\mu=0$.

 In the present paper, under some appropriate hypotheses on the behavior of
the potential of $f$, under a condition on the potential of $g$,
at infinity, we ensure the existence of infinitely many radially
symmetric solutions for the problem \eqref{1}; this is done in
Theorem \ref{thm2}. We also list some special cases of Theorem
\ref{thm2}. Further, replacing the conditions at infinity of the
potentials of $f$ and $g$, by a similar one at zero, the same
results hold and, in addition, the sequence of symmetric solutions
uniformly converges to zero; this is done in Theorem \ref{thm5}. The
abstract approach is fully based on the critical point theorem
proved in \cite{BM2}. Our approach here is in the one dimensional
setting and is different from that employed in \cite{K1} in which
the author directly discussed the existence of infinitely many
solutions for the original differential inclusion problem, while
here by setting $\rho = |x |$ and treating \eqref{1} as an
ordinary differential equation we establish the existence of
infinitely many solutions for the ordinary differential equation
which will be observed later (see \eqref{5}), and since the
solutions of the ordinary differential equation are the solutions
of the problem \eqref{1}, we have the results for the problem
\eqref{1}.

  A special case of our main result is the following theorem.

\begin{theorem}\label{thm1.1}
Let $f:\mathbb{R}^{N}\times\mathbb{R}\to
\mathbb{R}$ be continuous almost everywhere, and assume that
for each $\delta>0 $ there is a constant $M_{\delta}$ such that
$$
\sup_{| z|\leq \delta}| f(\rho, z )|\leq M_{\delta},
$$
where $\rho=| x|$, and that for all $z\in D(f)$ the condition
$f^{-}(\rho, z)\leq 0\leq f^{+}(\rho, z)$ implies $f(\rho, z)=0$,
 where
\begin{equation}\label{neweqf}
\begin{gathered}
f^{-}(\rho, z)={\lim_{\delta\to 0^{+}}}
\operatorname{ess\,inf}_{| z-\zeta | <\delta}f(\rho, \zeta),\\
f^{+}(\rho, z)={\lim_{\delta\to 0^{+}}}
\operatorname{ess\,sup}_{| z-\zeta | <\delta}f(\rho, \zeta).
\end{gathered}
\end{equation}
Put
$$
F(\rho,t)=\int^{t}_{0}f(\rho,s)ds,\quad \rho\in \mathbb{R}^{+}
\cup{\{0\}},\quad
t \in\mathbb{R}.
$$
Assume that
\begin{gather*}
\liminf _{\xi\to+\infty} \frac{\int_{0}^{+\infty}\sup
_{| t|\leq\xi} F(\rho,t) \rho^{N-1}d\rho}{\xi^{p}} =0,\\
\limsup_{\xi\to+\infty}\frac{\int_{0}^{\frac{D}{2}}
F(\rho,\xi)\rho^{N-1}d\rho}{\xi^p}=+\infty
\quad \text{for some } D>0.
\end{gather*}
Then, the problem
\[
-\Delta_{p}u+| u |^{p-2}u=f(|x|, u), \quad  x\in \mathbb{R}^{N},\quad
u \in W_{r}^{1,p}(\mathbb{R}^{N})
\]
admits a sequence of symmetric solutions.
\end{theorem}

\section{Basic definitions and preliminary results}

For basic notation and definitions on the subject,
we refer the reader to \cite{BM1,CLM,Molica,MR}.
 Let $(X,\|\cdot\|_X)$ be a real Banach space. We
denote by $X^\ast$ the dual space of $X$, while
$\langle\cdot,\cdot\rangle$ stands for the duality pairing between
$X^\ast$ and $X$. A function $\varphi:X\to\mathbb{R}$ is
called locally Lipschitz if, for all $u\in X$, there exist a
neighborhood $U$ of $u$ and a real number $L>0$ such that
$$
|\varphi(v)-\varphi(w)|\leq L\|v-w\|_X\quad\text{for all }
v,w\in U.
$$
If $\varphi$ is locally Lipschitz and $u\in X$,
the generalized directional derivative of $\varphi$ at $u$ along
the direction $v\in X$ is
$$
\varphi^\circ(u;v):=\limsup_{w\to u,\,\tau\to 0^+}
\frac{\varphi(w+\tau v)-\varphi(w)}{\tau}.
$$
The generalized gradient of $\varphi$ at $u$ is the set
$$
\partial\varphi(u):=\{u^\ast\in X^\ast :
\langle u^\ast,v\rangle\leq\varphi^\circ(u;v)\text{ for all }
v\in X\}.
$$
So $\partial\varphi:X\to 2^{X^\ast}$ is a
multifunction. We say that $\varphi$ has compact gradient if
$\partial\varphi$ maps bounded subsets of $X$ into relatively
compact subsets of $X^\ast$.


\begin{lemma}[{\cite[Proposition 1.1]{MotPan}}] \label{lem2.1}
Let $\varphi$ be a functional in $C^1(X)$. Then $\varphi$ is locally
Lipschitz and
\begin{gather*}
\varphi^\circ(u;v)=\langle\varphi'(u),v\rangle\quad\text{for all }u,v\in X;\\
\partial\varphi(u)=\{\varphi'(u)\}\quad\text{for all }u\in X.
\end{gather*}
\end{lemma}

\begin{lemma}[{\cite[Proposition 1.3]{MotPan}}] \label{lem2.2}
Let $\varphi:X\to\mathbb{R}$ be a locally Lipschitz
functional.
Then $\varphi^\circ(u;\cdot)$ is subadditive and
positively homogeneous for all $u\in X$, and
$$
\varphi^\circ(u;v)\leq L\|v\|\quad \text{for all }u,v\in X,
$$
with $L>0$ being a Lipschitz constant
for $\varphi$ with respect to $u$.
\end{lemma}


\begin{lemma}[\cite{Clarke}] \label{lem2.3}
Let $\varphi:X\to\mathbb{R}$ be a locally Lipschitz
functional. Then $\varphi^\circ:X\times X\to\mathbb{R}$ is
upper semicontinuous and for all $\lambda\geq 0$, $u,v\in X$,
$$
(\lambda\varphi)^\circ(u;v)=\lambda\varphi^\circ(u;v).
$$
Moreover, if $\varphi,\psi:X\to\mathbb{R}$ are locally Lipschitz
functionals, then
$$
(\varphi+\psi)^\circ(u;v)\leq\varphi^\circ(u;v)+\psi^\circ(u;v)\quad
\text{for all } u,v\in X.
$$
\end{lemma}

\begin{lemma}[{\cite[Proposition 1.6]{MotPan}}] \label{lem2.4}
Let $\varphi,\psi:X\to\mathbb{R}$ be locally Lipschitz
functionals. Then
$$
\partial(\lambda\varphi)(u)=\lambda\partial\varphi(u)\;\text{for
all}\;u\in X,\,\lambda\in\mathbb{R}, and
$$
$$
\partial(\varphi+\psi)(u)\subseteq\partial\varphi(u)+\partial\psi(u)\;\text{for
all}\;u\in X.
$$
\end{lemma}


 We say that $u\in X$ is a (generalized) critical point of a locally
Lipschitz functional $\varphi$ if $0\in\partial\varphi(u)$, i.e.,
$$
\varphi^\circ(u;v)\geq 0\quad\text{for all}\;v\in X.
$$
When a non-smooth functional, $g:X\to(-\infty,+\infty)$,
is expressed as a sum of a locally Lipschitz function,
$\varphi:X\to\mathbb{R}$, and a convex, proper, and lower
semicontinuous function, $j:X\to(-\infty,+\infty)$, that
is, $g:=\varphi+j$, a (generalized) critical point of $g$ is every
$u\in X$ such that
$$
\varphi^\circ(u;v-u)+j(v)-j(u)\geq 0
$$
for all $v\in X$ (see \cite[Chapter 3]{MotPan}).

 Let the space
$$
W^{1,p}(\mathbb{R}^{N})=\{ u\in L^{p}(\mathbb{R}^{N})
: \nabla u \in L^{p}(\mathbb{R}^{N})\},
$$
be equipped with the norm
$$
\| u\|_{W^{1,p}(\mathbb{R}^{N})}
=\Big(  \int_{\mathbb{R}^{N}}
(| \nabla u(x)|^{p}+| u(x)|^{p})dx\Big)^{1/p}.
$$
 The action of the orthogonal group $O(N)$ on $W^{1,p}(\mathbb{R}^{N})$
can be defined by $gu(x)=u(g^{-1}x)$ for every $g\in O(N)$,
  $u\in W^{1,p}(\mathbb{R}^{N})$ and
 $x \in \mathbb{R}^{N}$ (see \cite{W}), and we can
 define the subspace of radially symmetric functions of
 $W^{1,p}(\mathbb{R}^{N})$ by
 $$
W_{r}^{1,p}(\mathbb{R}^{N})
=\{ u \in W^{1,p}(\mathbb{R}^{N}): gu=u, \forall g \in O(N)\}
$$ 
equipped with the norm
$$
 \| u\|_{W_{r}^{1,p}(\mathbb{R}^{N})}
=\Big(\int_{0}^{+\infty}(|u'(\rho)|^{p}+| u(\rho)
|^p)\rho^{N-1}d\rho \Big)^{1/p}.
$$
  As pointed out in \cite[Theorem 3.1]{K1}, since $2\leq
N<p<+\infty$, $W_{r}^{1,p}(\mathbb{R}^{N})$ is compactly embedded
in $L^\infty(\mathbb{R}^{N})$. In particular, there exists a
positive constant $k>0$ such that
\begin{equation}\label{2}
\sup_{\rho\in[0,+\infty]}|u(\rho)|\leq k\|
u\|_{W_{r}^{1,p}(\mathbb{R}^{N})}
\end{equation} for each $u\in
W_{r}^{1,p}(\mathbb{R}^{N})$.

 Hereafter, we assume that $X$ is a reflexive real Banach space,
$\Phi:X\to\mathbb{R}$ is a sequentially weakly
lower semicontinuous functional, $\Upsilon:X\to\mathbb{R}$
is a sequentially weakly upper semicontinuous functional,
$\lambda$ is a positive parameter,
$j:X\to(-\infty,+\infty)$ is a convex, proper, and lower
semicontinuous functional, and $D(j)$ is the effective domain of
$j$.
Write
$$
\Psi:=\Upsilon-j,\qquad\qquad I_\lambda:=\Phi-\lambda\Psi
=(\Phi-\lambda\Upsilon)+\lambda j.
$$
We also assume that $\Phi$ is coercive and
\begin{equation}\label{3}
D(j)\cap\Phi^{-1}(-\infty,r)\neq\emptyset
\end{equation}
for all $r>\inf_X\Phi$. Moreover, owing to \eqref{3} and provided
$r>\inf_X\Phi$, we can define
\begin{gather*}
\varphi(r):=\inf_{u\in\Phi^{-1}(-\infty,r)}
\frac{\big(\sup_{v\in\Phi^{-1}(-\infty,r)}\Psi(v)\big)-\Psi(u)}{r-\Phi(u)},
\\
\gamma:=\liminf_{r\to
+\infty}\varphi(r),\quad
\delta:=\liminf_{r\to(\inf_X\Phi)^+}\varphi(r).
\end{gather*}
When $\Phi$ and $\Upsilon$ are locally Lipschitz functionals
the following result is proved in \cite[Theorem 2.1]{BM2}; it is a
more precise version of \cite[Theorem 1.1]{MM} (see also
\cite{R}), which is the main tool to prove our results.

\begin{theorem}\label{thm1}
Under the above assumptions on $X,\Phi$ and $\Psi$, one has

{\rm (a)} For every $r>\inf_X\Phi$ and every $\lambda\in
(0,1/\varphi(r))$, the restriction of the functional
$I_\lambda=\Phi-\lambda\Psi$ to $\Phi^{-1}(-\infty,r)$ admits a
global minimum, which is a critical point (local minimum) of
$I_\lambda$ in $X$.

{\rm (b)} If $\gamma<+\infty$, then for
each $\lambda\in (0,1/\gamma)$, the following alternative holds:
either
\begin{itemize}
\item[(b1)] $I_\lambda$ possesses a global minimum, or

\item[(b2)] there is a sequence $\{u_n\}$ of critical points
(local minima) of $I_\lambda$ such that
$\lim_{n\to+\infty}\Phi(u_n)=+\infty$.
\end{itemize}

{\rm (c)} If $\delta<+\infty$, then for each $\lambda\in
(0,1/\delta)$, the following alternative holds: either
\begin{itemize}
\item[(c1)] there is a global minimum of $\Phi$ which
is a local minimum of $I_\lambda$, or
\item[(c2)] there is a sequence $\{u_n\}$ of pairwise distinct critical points (local
minima) of $I_\lambda$, with
$\lim_{n\to+\infty}\Phi(u_n)=\inf_X\Phi$,
which weakly converges to a global minimum of $\Phi$.
\end{itemize}
\end{theorem}

\section{Main results}

Let $f:\mathbb{R}^{N}\times\mathbb{R}\to \mathbb{R}$ be continuous
almost everywhere and assume that
for each $\delta_1>0 $ there is a constant $ M_{\delta_1}$ such that
\begin{equation}\label{4}\sup_{| z|\leq \delta_1}|
f(| x|, z )|\leq M_{\delta_1}.
\end{equation}

 Since $x$ is away from the origin, we set $\rho = |x |$ and treat
\eqref{1} as an ordinary differential equation. Thus we write
$u(\rho)$ instead of $u(x )$, and the problem \eqref{1}
corresponds exactly to
\begin{equation}\label{5}
-(\rho^{N-1}\phi(u'))'+\rho^{N-1}\phi(u)=\lambda
\rho^{N-1}f(\rho,u)+\mu \rho^{N-1}g(\rho,u)
\end{equation}
where
$'$ denotes $ \frac{d}{d\rho}$ and $\phi(s)=| s|^{p-2}s$.
Put
$$
F(\rho,t)=\int^{t}_{0}f(\rho,s)ds,\ \rho\in \mathbb{R}^{+} \cup{\{0\}},
\quad t \in\mathbb{R}.
$$
Pick $D>0$ such that $S(0,D)\subseteq  \mathbb{R}^{N}$ where
$S(0, D)$ denotes the ball with center at $0$ and radius of $D$, and let
$\omega_N$ be the volume of the $N$-dimensional unit ball.

Our main result is stated using the following assumptions:
\begin{itemize}
\item[(A1)] $F(\rho,t)\geq 0$ for all $(\rho,t)
\in[\frac{D}{2},+\infty)\times(\mathbb{R}^+\cup\{0\})$;

\item[(A2)]
\begin{align*}
&\liminf _{\xi\to+\infty} \frac{\int_{0}^{+\infty}
(\sup _{| t|\leq\xi} F(\rho,t)) \rho^{N-1}d\rho}{\xi^{p}} \\
&< \frac{1}{k^p\omega_{N} D^{N}(\frac{2^p}{D^p}(1-\frac{1}{2^N})+1)}
\limsup_{\xi\to+\infty}\frac{\int_{0}^{\frac{D}{2}}F(\rho,\xi)\rho^{N-1}d\rho
}{\xi^p}\,;
\end{align*}

\item[(A3)] for all $z\in D(f)$ the condition $f^{-}(\rho, z)\leq 0\leq
f^{+}(\rho, z)$ implies $f(\rho, z)=0$, where $f^{-}(\rho, z)$ and
$f^{+}(\rho, z)$ are given as in \eqref{neweqf}.

Put
\begin{gather*}
\lambda_1 : =\frac{\omega_{N}
D^{N}(\frac{2^p}{D^p}(1-\frac{1}{2^N})+1)}{
p\limsup_{\xi\to+\infty}\frac{\int_{0}^{\frac{D}{2}}F(\rho,\xi)\rho^{N-1}d\rho
}{\xi^p}}
\\
\lambda_2:=\Big(pk^p\liminf _{\xi\to+\infty}
\frac{\int_{0}^{+\infty}(\sup _{| t|\leq\xi} F(\rho,t))
\rho^{N-1}d\rho}{\xi^{p}} \Big)^{-1}.
\end{gather*}
Suppose that
$g:\mathbb{R}^{N}\times\mathbb{R}\to \mathbb{R}$ is continuous
almost everywhere, and for $\delta_2 >0 $ there
is a constant $M_{\delta_2}$ such that
\begin{equation}\label{6}
\sup_{| z|\leq \delta_{2}}| g(| x|, z
)|\leq M_{\delta_{2}},
\end{equation}

\item[(A4)]
for all $z\in D(g)$ the condition $g^{-}(\rho, z)\leq
0\leq g^{+}(\rho, z)$ implies $g(\rho, z)=0$,
 where 
\[
g^{-}(\rho, z)=\lim_{\delta\to 0^{+}} 
\operatorname{ess\,inf}_{| z-\zeta | <\delta}g(\rho, \zeta),\quad
g^{+}(\rho, z)=\lim_{\delta\to 0^{+}} \operatorname{ess\,sup}_{| z-\zeta | 
<\delta}g(\rho,  \zeta),
\]
whose potential $G(\rho,t)=\int^{t}_{0}g(\rho,s)ds$, 
$\rho\in \mathbb{R}^{+} \cup{\{0\}}$, $t \in\mathbb{R}$, is a
non-negative function satisfying the condition
\begin{equation}\label{7}
 g_\infty:=\lim_{\xi\to+\infty}\frac{\int_{0}^{+\infty}(\sup_{|
t|\leq\xi}G(\rho,t)) \rho^{N-1}d\rho}{\xi^p}<+\infty\,.
\end{equation}
Set
$$
\mu_{g,\lambda}:=\frac{1}{pk^{p} g_{\infty}}
\Big(1-\lambda p{k}^p  \liminf_{\xi\to
+\infty}\frac{\int_{0}^{+\infty}(\sup_{|
t|\leq\xi}F(\rho,t))\rho^{N-1}d\rho}{\xi^{p}}\Big).
$$
\end{itemize}

\begin{theorem}\label{thm2}
Under assumptions {\rm (A1)--(A4)}, 
for each $\lambda\in]\lambda_1,\lambda_2[$ and for every
$\mu\in[0,\mu_{g,\lambda}[$, problem \eqref{1} has an unbounded
sequence of symmetric solutions.
\end{theorem}

\begin{proof}
To apply Theorem \ref{thm1} to our problem, we take 
$X=W_{r}^{1,p}(\mathbb{R}^{N})$. Fix
$\overline{\lambda}\in]\lambda_1,\lambda_2[$ and let $g$ be an
almost everywhere continuous function satisfying the condition
\eqref{7}. Arguing as in \cite{BM1}, we follow the proof in the
case $\mu>0$. Since, $\overline{\lambda}<\lambda_2$, one has
$$
\mu_{g,\overline {\lambda}}:=\frac{1}{pk^p g_\infty} 
\Big(1-\overline{\lambda}\ p{k}^p \liminf_{\xi\to+\infty}
\frac{\int_{0}^{+\infty} (\sup_{| t|\leq\xi}
F(\rho,t))\rho^{N-1}d\rho}{\xi^{p}}\Big)>0.
$$ 
Fix $\overline{\mu}\in]0,\mu_{g,\overline{\lambda}}[$ and set
$\nu_1:=\lambda_1$ and
$\nu_2:=\frac{\lambda_2}{1+p{k}^p
\frac{\overline{\mu}}{\overline{\lambda}}\lambda_2g_\infty}$.
If $g_\infty=0$, clearly, $\nu_1=\lambda_1$, $\nu_2=\lambda_2$ and
$\lambda\in]\nu_1,\nu_2[$. If $g_\infty\neq0$, since
$\overline{\mu}<\mu_{g,\overline{\lambda}}$, we obtain
$$
\frac{\overline{\lambda}}{\lambda_2}+p{k}^p \overline{\mu}g_\infty<1,
$$
and so
$$
\frac{\lambda_2}{1+pk^p \frac{\overline{\mu}}{\overline{\lambda}}
\lambda_2g_\infty}>\overline{\lambda},
$$
namely, $\overline{\lambda}<\nu_2$. 
Hence, since
$\overline{\lambda}>\lambda_1=\nu_1$, one has
$\overline{\lambda}\in]\nu_1,\nu_2[$. We now set
\begin{gather*}
\Phi(u)=\frac{1}{p}\|u\|^{p}_{ W_{r}^{1,p}(\mathbb{R}^{N})},\quad
\Upsilon(u)=\int_{0}^{+\infty}[F(\rho,u)+\frac{\overline{\mu}}
{\overline{\lambda}}G(\rho,u)]\rho^{N-1}d\rho,\\
j(u)=0, \quad
\Psi(u)=\Upsilon(u)-j(u)=\Upsilon(u)
\end{gather*}
for each $u\in X$.
Clearly, the functional $\Phi$ is locally Lipschitz and weakly
sequentially lower semi-continuous. Put
$I_{\overline{\lambda}}:=\Phi-\overline{\lambda}\Psi$. 
Since  $f$ and $g$ satisfy \eqref{4} and \eqref{6},
respectively, and $W_{r}^{1,p}(\mathbb{R}^{N})$ is compactly
embedded in $L^{\infty} (\mathbb{R}^{N})$, the assertion remains
true regarding $\Psi $ too (see \cite{K2,KV}). By a simple
computation, we obtain
$$
\frac{d\Phi(u)}{du}=\int_{0}^{+\infty}[-(|
u'|^{p-2}u')'+| u|^{p-2}u]\rho^{N-1}d\rho.
$$ 
From  Chang \cite[Theorem 2.1]{Ch}, we have
$$
\partial\Psi(u)=[(f^{-}(\rho,u)+\frac{\overline{\mu}}{\overline{\lambda}}g^{-}
(\rho,u))\rho^{N-1},
(f^{+}(\rho,u)+\frac{\overline{\mu}}{\overline{\lambda}}g^{+}(\rho,u))\rho^{N-1}].
$$
So the critical point of the functional $I_{\overline{\lambda}}$
is precisely the solution of the differential inclusion
\begin{equation} \label{8}
\begin{aligned}
&-(\rho^{N-1}\phi(u'))'+\rho^{N-1}\phi(u) \\
&\in \overline{\lambda} [(f^{-}(\rho,u)
+\frac{\overline{\mu}}{\overline{\lambda}}g^{-}
(\rho,u))\rho^{N-1}, (f^{+}(\rho,u)
+\frac{\overline{\mu}}{\overline{\lambda}}g^{+}(\rho,u))\rho^{N-1}]
\end{aligned}
\end{equation} 
for $\rho \in [0,+\infty]\backslash (u^{-1} (D_{f})\bigcup u^{-1}
(D_{g}))$.

Since $m(D_{f})=m(D_{g})=0$, we can obtain
$-(\rho^{N-1}\phi(u'))'+\rho^{N-1}\phi(u))=0$ for almost all $\rho
\in u^{-1} (D_{f})\cap u^{-1} (D_{g})$. On the other hand, in
view of Assumptions (A3) and (A4), we obtain $f(\rho,u(\rho))=0$
for almost all $ \rho \in u^{-1}(D_{f})$
 and $g(\rho, u(\rho))=0$ for almost all $ \rho \in u^{-1}(D_{g})$,
 respectively, i.e.
  \begin{equation} \label{9}
 -(\rho^{N-1}\phi(u'))'+\rho^{N-1}\phi(u)=\overline{\lambda}\rho^{N-1}f(\rho,u)
  +\overline{\mu}\rho^{N-1}g(\rho,u)
\end{equation}
for almost all $\rho \in u^{-1}(D_{f})\cap u^{-1} (D_{g})$. 
Combining \eqref{8} and \eqref{9}, we can obtain that the
solutions of the problem \eqref{5} are exactly the critical points
of the functional $I_{\overline{\lambda}}$. Now, we claim that
$\gamma<+\infty$.

Let $\{\xi_n\}$ be a sequence of positive
numbers such that $\xi_n\to+\infty$ as $n\to\infty$ and
\begin{align*}
&\lim_{n\to \infty}\frac{\int_{0}^{+\infty} (\sup_{|
t|\leq\xi_{n}}[F(\rho,t)
+\frac{\overline{\mu}}{\overline{\lambda}}G(\rho,t)])\rho^{N-1}d\rho}{\xi_n^{p}}\\
&=\liminf_{\xi\to +\infty}\frac{\int_{0}^{+\infty} (\sup_{| t|\leq\xi}
[F(\rho,t)+\frac{\overline{\mu}}{\overline{\lambda}}G(\rho,t)])
\rho^{N-1}d\rho}{\xi^{p}}.
\end{align*}
Put $r_n=\frac{1}{p}(\frac{
\xi_n}{k})^{p}$ for all $n\in \mathbb{N}$. Bearing  in
mind \eqref{2}, we have
\begin{align*}
\Phi^{-1}(-\infty,r_{n})
&= \{ u\in X; \Phi(u)< r_{n}\}\\
&= \{ u\in X; \|u\|^p_{W_{r}^{1,p}(\mathbb{R}^{N})}< pr_{n}\}\\
& \subseteq \{ u\in X; |u(\rho)|\leq \xi_n\ \text{for all}\ \rho\in
[0,+\infty]\}.
\end{align*}
Hence, taking into account that
$ \inf_{X}\Phi(0)=0$ and $\Psi(0)=0$ for every $n$ large enough,
 one has
\begin{align*}
\varphi(r_n)
&= \inf_{u\in\Phi^{-1}(-\infty,r_n)}\frac{(\sup_{v\in\Phi^{-1}(-\infty,r_n)}
\Psi(v))-\Psi(u)}{r_n-\Phi(u)}\\
&\leq \frac{\sup_{v\in\Phi^{-1}(-\infty,r_n)}\Psi(v)}{r_n}\\
&\leq \frac{\int_{0}^{+\infty} (\sup_{|
t|\leq\xi_{n}}\Big{[}F(\rho,t)+\frac{\overline{\mu}}{\overline{\lambda}}G(\rho,t)
\Big{]})\rho^{N-1}d\rho}{\frac{1}{p}({\frac{
\xi_n}{k}})^{p}}\\
&\leq \frac{\int_{0}^{+\infty}(\sup_{|
 t|\leq\xi_{n}}F(\rho,t))\rho^{N-1}d\rho}{\frac{1}{p}
 ({\frac{ \xi_n}{k}})^{p}}
+\frac{\overline{\mu}}{\overline{\lambda}}
 \frac{\int_{0}^{+\infty}(\sup_{|
 t|\leq\xi_{n}}G(\rho,t))\rho^{N-1}d\rho}{\frac{1}{p}
 ({\frac{ \xi_n}{k}})^{p}}.
\end{align*}
From Assumption (A2) and the condition \eqref{7} one has
\begin{align*}
&\lim_{n\to \infty}\frac{\int_{0}^{+\infty}(\sup_{|
t|\leq\xi_{n}}
F(\rho,t))\rho^{N-1}d\rho}{\frac{1}{p}({\frac{
\xi_n}{k}})^{p}}
+\lim_{n\to \infty}\frac{\overline{\mu}}{\overline{\lambda}}
\frac{\int_{0}^{+\infty}(\sup_{|
t|\leq\xi_{n}}G(\rho,t))\rho^{N-1}d\rho}{\frac{1}{p}({\frac{
\xi_n}{k}})^{p}}\\
&<+\infty,
\end{align*}
from which it follows that
$$
\lim_{n\to\infty}\frac{\int_{0}^{+\infty} (\sup_{| t|\leq\xi_{n}}
[F(\rho,t)+\frac{\overline{\mu}}{\overline{\lambda}}G(\rho,t)])
\rho^{N-1}d\rho}{\frac{1}{p}({\frac{
\xi_n}{k}})^{p}} <+\infty.
$$ 
Therefore,
\begin{equation}\label{10}
\gamma\leq \liminf_{n\to+\infty}\varphi(r_n)\leq\lim_{n\to
\infty}\frac{\int_{0}^{+\infty} (\sup_{|
t|\leq\xi_{n}}[F(\rho,t)
+\frac{\overline{\mu}}{\overline{\lambda}}G(\rho,t)])\rho^{N-1}d\rho}
{\frac{1}{p}(\frac{\xi_n}{k})^{p}}<+\infty.
\end{equation}
Since
\begin{align*}
&\frac{\int_{0}^{+\infty}(\sup_{| t|\leq\xi}
[F(\rho,t)+\frac{\overline{\mu}}{\overline{\lambda}}G(\rho,t)])\rho^{N-1}d\rho}
{\frac{1}{p}(\frac{\xi}{k})^{p}}\\
&\leq\frac{\int_{0}^{+\infty}(\sup_{| t|\leq\xi}F(\rho,t))\rho^{N-1}d\rho}
{\frac{1}{p}({\frac{ \xi}{k}})^{p}}
+\frac{\overline{\mu}}{\overline{\lambda}}
\frac{\int_{0}^{+\infty}(\sup_{| t|\leq\xi} G(\rho,t))
\rho^{N-1}d\rho}{\frac{1}{p}({\frac{\xi}{k}})^{p}},
\end{align*}
taking  into account \eqref{7}, one has
\begin{equation}\label{11}
\begin{aligned}
&\liminf_{\xi\to +\infty}\frac{\int_{0}^{+\infty}(\sup_{|
t|\leq\xi}[F(\rho,t)
+\frac{\overline{\mu}}{\overline{\lambda}}G(\rho,t)])\rho^{N-1}d\rho}{\xi^{p}}\\
&\leq\liminf_{\xi\to +\infty}\frac{\int_{0}^{+\infty}(\sup_{|
t|\leq\xi}F(\rho,t))\rho^{N-1}d\rho}{\xi^{p}}
+\frac{\overline{\mu}}{\overline{\lambda}}g_\infty.
\end{aligned}
 \end{equation}
Since $G$ is nonnegative, we obtain
\begin{equation} \label{12}
\limsup_{\xi\to+\infty}\frac{\int_{0}^{\frac{D}{2}}[F(\rho,\xi)
+\frac{\overline{\mu}}{\overline{\lambda}}G(\rho,\xi)]\rho^{N-1}d\rho
}{\xi^p}\geq
\limsup_{\xi\to+\infty}\frac{\int_{0}^{\frac{D}{2}}F(\rho,\xi)\rho^{N-1}d\rho
}{\xi^p}.
\end{equation}
Therefore, in view of \eqref{11} and \eqref{12}, we have
\begin{align*}
\overline {\lambda}
&\in]\nu_1,\nu_2[\\
&\subseteq \Big]\frac{\omega_{N}
D^{N}(\frac{2^p}{D^p}(1-\frac{1}{2^N})+1)}{
p\limsup_{\xi\to+\infty}\frac{\int_{0}^{\frac{D}{2}}[F(\rho,\xi)
+\frac{\overline{\mu}}{\overline{\lambda}}G(\rho,\xi)]\rho^{N-1}d\rho
}{\xi^p}} , \\
&\quad \frac{1}{pk^p\liminf _{\xi\to+\infty}
\frac{\int_{0}^{+\infty}(\sup _{| t|\leq\xi} [F(\rho,t)
+\frac{\overline{\mu}}{\overline{\lambda}}G(\rho,t)])
\rho^{N-1}d\rho}{\xi^{p}} }\Big[\\
&\subseteq ]0,\frac{1}{\gamma}[,
\end{align*} 
where we used (A2) and \eqref{10}.

For a  fixed $\overline{\lambda}$,  inequality \eqref{10} implies that the
condition (b) of Theorem \ref{thm1} can be applied and either
$I_{\overline{\lambda}}$ has a global minimum or there exists a
sequence $\{u_n\}$ of solutions of the problem \eqref{5} such
that $\lim_{n\to\infty}\| u_{n} \|=+\infty$.

The other step is to show that for the fixed $\overline{\lambda}$
the functional $I_{\overline{\lambda}}$ has no global minimum. Let
us verify that the functional $I_{\overline{\lambda}}$ is
unbounded from below. Since
\begin{align*}
\frac{1}{{\overline{\lambda}}}
&< \frac{p}{\omega_{N}D^{N}(\frac{2^p}{D^p}(1-\frac{1}{2^N})+1)}
\limsup_{\xi\to+\infty}
\frac{\int_{0}^{\frac{D}{2}}F(\rho,\xi)\rho^{N-1}d\rho }{\xi^p}\\
&\leq
 \frac{p}{\omega_{N}D^{N}(\frac{2^p}{D^p}(1-\frac{1}{2^N})+1)}
\limsup_{\xi\to +\infty}
 \frac{\int_{0}^{\frac{D}{2}}[F(\rho,\xi)
 +\frac{\overline{\mu}}{\overline{\lambda}}G(\rho,\xi)]\rho^{N-1}d\rho}{\xi^{p}},
\end{align*}
we can consider a real sequence $\{d_n\}$ and a positive constant
$\tau$ such that $d_n\to +\infty$ as $n\to \infty$ and
\begin{equation}\label{13}
\frac{1}{{\overline{\lambda}}}<\tau
< \frac{p}{\omega_{N}D^{N}(\frac{2^p}
{D^p}(1-\frac{1}{2^N})+1)}\frac{\int_{0}^{\frac{D}{2}}[F(\rho,d_n)
+\frac{\overline{\mu}}{\overline{\lambda}}G(\rho,d_n)]
\rho^{N-1}d\rho}{d_{n}}
\end{equation}
for each $n\in\mathbb{N}$ large enough. Let $\{w_n\}$ be a sequence 
in $X$ defined by
\begin{equation}\label{14}
w_{n}(x)= \begin{cases}
0 &\text{if } x\in\mathbb{R}^{N} \setminus S(0,D), \\
\frac{2d_{n}}{D}(D-| x |) &\text{if } x\in S(0,D)
\setminus S(0,\frac{D}{2}),\\ 
{d_{n}} &\text{if }  x\in S(0,\frac{D}{2}).
\end{cases}
\end{equation}
For any fixed $n\in \mathbb{N}$, it is easy to see that 
$w_{n}\in X$ and, in particular, one has
\begin{equation}\label{15}
\|w_{n} \|_{ W_{r}^{1,p}(\mathbb{R}^{N})}^{p} 
\leq d_{n}^p \omega_{N} D^N\big(\frac{2^p}{D^p}(1-\frac{1}{2^N})+1\big).
\end{equation} 
On the other hand, since $0\leq w_n(x)\leq d_n$ for every
$x\in\mathbb{R}^N$, from (A1) and since $G$ is nonnegative, from
the definition of $\Psi$, we infer
\begin{equation}\label{16}
\Psi(w_n)\geq\int_{0}^{\frac{D}{2}}[F(\rho,d_n)
+\frac{\overline{\mu}}{\overline{\lambda}}G(\rho,d_n)]\rho^{N-1}d\rho.
\end{equation}
So, according to \eqref{13}, \eqref{15} and \eqref{16}, we obtain
\begin{align*}
I_{\overline{\lambda}}(w_{n})
&\leq\frac{1}{p} d_{n}^p \omega_{N}D^N\big(\frac{2^p}{D^p}(1-\frac{1}{2^N})+1\big)
 -{\overline{\lambda}}\int_{0}^{\frac{D}{2}}[F(\rho,d_n)
+\frac{\overline{\mu}}{\overline{\lambda}}G(\rho,d_n)]
  \rho^{N-1}d\rho\\
&<\frac{1}{p}d_{n}^p \omega_{N}D^N\big(\frac{2^p}{D^p}(1-\frac{1}{2^N})+1\big)
(1-{\overline{\lambda}}\tau)
\end{align*}
for every $n\in \mathbb{N}$ large enough.
 Since  $ {\overline{\lambda}}\tau >1 $ and
$\lim_{n\to+\infty} d_n =+\infty$ we have
$$
\lim_{n\to+\infty}I_{\overline{\lambda}}(w_{n})=-\infty\,.
$$ 
Hence, the functional $I_{\overline{\lambda}}$ is unbounded from below, 
and it follows that $I_{\overline{\lambda}}$ has no global minimum. Therefore,
applying Theorem \ref{thm1} we deduce that there is a sequence
$\{u_n\}\subset X$ of critical points of $I_{\overline{\lambda}}$
such that $\lim_{n\to\infty}\| u_{n}\|_{
W_{r}^{1,p}(\mathbb{R}^{N})}=+\infty$.
Hence, since the critical
points of the functional $I_{\overline{\lambda}}$ are exactly the
solutions of the problem \eqref{5}, and then they are the
solutions of the problem \eqref{1}, the conclusion is achieved.
\end{proof}

\begin{remark}\label{rmk1} \rm
We notice that instead of Assumption (A2) in Theorem
\ref{thm2} we are allowed to assume the  more general condition
\begin{itemize}
\item[(A5)] there exist two sequence $\{\alpha_n\}$ and $\{\beta_n\}$
with 
\[
\Big(\omega_{N}D^N(\frac{2^p}{D^p}
 (1-\frac{1}{2^N})+1)\Big)^{1/p}\alpha_n <\frac{\beta_n}{k}
\]
for every $n\in \mathbb{N}$ and $\lim_{n\to+\infty}\beta_n=+\infty$ such that
\begin{align*}
&\lim_{n\to +\infty}\frac{\int_{0}^{+\infty}
 (\sup_{| t|\leq\beta_n}F(\rho,t)) \rho^{N-1} d\rho-
\int_{0}^{\frac{D}{2}}F(\rho,\alpha_{n})\rho^{N-1}d\rho}{({{\frac{
\beta_n}{k}})^{p}
-\omega_{N}D^N(\frac{2^p}{D^p}(1-\frac{1}{2^N})+1)}
\alpha_{n}^p}\\
&<\frac{1}{\omega_{N} D^N (\frac{2^p}{D^p}
(1-\frac{1}{2^N})+1)}\limsup_{\xi\to
+\infty}\frac{\int_{0}^{\frac{D}{2}}F(\rho,\xi)\rho^{N-1}d\rho}{
\xi^p}. 
\end{align*}
\end{itemize}
Obviously, Assumption (A2) follows from Assumption
(A5), by choosing $\alpha_n=0$ for all $n\in\mathbb{N}$. Moreover,
if we assume (A5) instead of (A2) and set
$r_n=\frac{1}{p}(\frac{ \beta_n}{k})^{p}$ for all $n\in
\mathbb{N}$, by the same reasoning as in Theorem \ref{thm2}, we
obtain
\begin{align*}
\varphi(r_n)
&= \inf_{u\in\Phi^{-1}(-\infty,r_n)}\frac{(\sup_{v\in\Phi^{-1}(-\infty,r_n)}\Psi(v))
-\Psi(u)}{r_n-\Phi(u)}\\
&\leq \frac{\sup_{v\in\Phi^{-1}(-\infty,r_n)}\Psi(v)-\int_{0}^{+\infty}
F(\rho,w_{n}(x))\rho^{N-1}d\rho}{r_n-\frac{1}{p}\|
w_{n}\|_{ W_{r}^{1,p}(\mathbb{R}^{N})}^p }\\
&\leq \frac{\int_{0}^{+\infty}(\sup_{| t |
\leq\in\xi}F(\rho,t))\rho^{N-1}d\rho-\int_{0}^{\frac{D}{2}}
F(\rho,\alpha_n)\rho^{N-1}d\rho}{\frac{1}{p}({{\frac{
\beta_n}{k}})^{p}-\frac{1}{p}\omega_{N}
D^N(\frac{2^p}{D^p}(1-\frac{1}{2^N})+1)\alpha_n^p}},
\end{align*}
where $w_n(x)$ is defined as given in \eqref{14}, for
$x\in\mathbb{R}^N$ with $\alpha_n$ instead of $d_n$. We then have
the same conclusion as in Theorem \ref{thm2} with $\lambda_2$
replaced by
$$
\lambda'_{2}:=\Big(p\lim_{n\to
+\infty}\frac{\int_{0}^{+\infty}(\sup_{|
t|\leq\xi}F(\rho,t))\rho^{N-1} d\rho-
\int_{0}^{\frac{D}{2}}F(\rho,\alpha_{n})\rho^{N-1}d\rho}{({{\frac{\beta_n}{k}})^{p}
-\omega_{N}D^N(\frac{2^p}{D^p}(1-\frac{1}{2^N})+1)}
\alpha_{n}^p}\Big)^{-1}
$$
\end{remark}

The following result is a special case of Theorem \ref{thm2} with
$\mu=0$.

\begin{theorem}\label{thm3}
Assume that {\rm (A1)--(A3)} hold.
Then, for each
\begin{align*}
\lambda\in\Lambda_1
&:=\Big]\frac{\omega_{N}
D^{N}(\frac{2^p}{D^p}(1-\frac{1}{2^N})+1)}{
p\limsup_{\xi\to+\infty}\frac{\int_{0}^{\frac{D}{2}}F(\rho,\xi)\rho^{N-1}d\rho
}{\xi^p}},\\
&\quad  \frac{1}{pk^p\liminf _{\xi\to+\infty}
\frac{\int_{0}^{+\infty}(\sup _{| t|\leq\xi} F(\rho,t))
\rho^{N-1}d\rho}{\xi^{p}} }\Big[
\end{align*}
the problem 
\begin{equation} \label{17} 
-\Delta_{p}u+| u |^{p-2}u=\lambda f(| x|, u),
 \quad  x\in \mathbb{R}^{N}, \quad u\in W_{r}^{1,p}( \mathbb{R}^{N})
\end{equation} 
has an unbounded sequence of symmetric solutions.
\end{theorem}

 Here we point out the following consequence of Theorem \ref{thm3}.

\begin{corollary}\label{c1} 
Assume that {\rm (A1)} and {\rm(A3)} hold. Also assume that:
\begin{itemize}
\item[(A6)]  
\[
\liminf_{\xi\to +\infty}\frac{\int_{0}^{+\infty}(\sup_{| t|\leq\xi}
F(\rho,t))\rho^{N-1}d\rho}{\xi^{p}}<\frac{1}{pk^p};
\]
\item[(A7)] 
\[
\limsup_{\xi\to
+\infty}\frac{\int_{0}^{D/2}F(\rho,\xi)\rho^{N-1}d\rho}
{\xi^p}>\frac{1}{p}\omega_{N} D^N(\frac{2^p}{D^p}(1-\frac{1}{2^N})+1) \,.
\]
\end{itemize}
Then the problem 
\[
 -\Delta_{p}u+| u |^{p-2}u= f(| x|, u),
 \quad  x\in \mathbb{R}^{N},\quad u\in W_{r}^{1,p}( \mathbb{R}^{N})
\]
 has an unbounded sequence of symmetric
solutions. 
\end{corollary}

\begin{remark}\label{rmk2}\rm
Theorem \ref{thm1.1}  is an immediately
consequence of Corollary \ref{c1}.
\end{remark}

Now, we point out a special situation of Theorem \ref{thm3} when the
nonlinear term has separable variables. To be precise, let
$\alpha$ be a continuous function such that $\alpha(|x|)\geq 0$
a.e. $x\in \mathbb{R}^{N}$, $\alpha\not\equiv 0$, and let
$h:\mathbb{R}\to\mathbb{R}$ be non-negative and continuous
almost everywhere; namely, 
$m(D_{h})=0$ where
$D_{h}= \{ z\in \mathbb{R}, h(z) \text{ is discontinuous
at }  z\}$. 
We also assume that for each $ \iota>0 $ there is a constant
 $M_{\iota}$ such that 
$$
\sup_{| z|\leq \iota}| h( z )|\leq M_{\iota}. 
$$ 
Put
$H(t)=\int^{t}_{0}h(s)ds$, $t \in\mathbb{R}$. Then, we have the
following consequence of Theorem \ref{thm2}.

\begin{theorem}\label{thm4} 
Assume that
\begin{itemize}
\item[(A8)]
\[
\liminf _{\xi\to+\infty} \frac{ H(\xi) }{\xi^{p}} <
\frac{\int_{0}^{\frac{D}{2}}\alpha(\rho)\rho^{N-1}d\rho}{k^p\omega_{N}
D^{N}(\frac{2^p}{D^p}(1-\frac{1}{2^N})+1)(
\int_{0}^{+\infty}\alpha(\rho)\rho^{N-1}d\rho)}
\limsup_{\xi\to+\infty}\frac{H(\xi)}{\xi^p};
\]

\item[(A9)] for all $z\in D(h)$ the condition $h^{-}( z)\leq 0\leq h^{+}(
z)$ implies $h(z)=0$,
 where 
\[
h^{-}(z)={\lim_{\delta\to 0^{+}}}\operatorname{ess\,inf}_{| z-\zeta | 
<\delta}h(\zeta),\quad
h^{+}(z)={\lim_{\delta\to 0^{+}}}\operatorname{ess\,sup}_{| z-\zeta |
 <\delta}h(\zeta).
\]
\end{itemize}
Put
\begin{align*}
\Lambda_2&:=\Big{]}\frac{\omega_{N}
D^{N}(\frac{2^p}{D^p}(1-\frac{1}{2^N})+1)}{
p(\int_{0}^{\frac{D}{2}}\alpha(\rho)\rho^{N-1}d\rho)
\limsup_{\xi\to+\infty}\frac{H(\xi) }{\xi^p}}, \\
&\quad \frac{1}{pk^p(\int_{0}^{+\infty}\alpha(\rho)\rho^{N-1}d\rho)
\liminf_{\xi\to+\infty}
\frac{ H(\xi)}{\xi^{p}} }\Big{[}.
\end{align*}
Suppose that
$g:\mathbb{R}^{N}\times\mathbb{R}\to \mathbb{R}$ is an almost
everywhere continuous function such that for $\delta_2 >0 $ there
is a constant $M_{\delta_2}$ such that \eqref{6} holds, and
satisfies (A4), whose potential
$G(\rho,t)=\int^{t}_{0}g(\rho,s)ds, \ \rho\in \mathbb{R}^{+}
\cup{\{0\}}, \ t \in\mathbb{R}$, is a non-negative
function satisfying the condition \eqref{7}.
Set
$$
\mu'_{g,\lambda}:=\frac{1}{pk^{p} g_{\infty}}\Big(1-\lambda p{k}^p (
\int_{0}^{+\infty}\alpha(\rho)\rho^{N-1}d\rho)\liminf
_{\xi\to+\infty} \frac{ H(\xi) }{\xi^{p}}\Big).
$$
Then, for each $\lambda\in\Lambda_2$ and for every
$\mu\in[0,\mu'_{g,\lambda}[$ the problem
\begin{equation}\label{18}
 -\Delta_{p}u+| u |^{p-2}u=\lambda \alpha(| x|)h(u)+\mu g(| x|,u),
 \quad  x\in \mathbb{R}^{N}, \quad u\in W_{r}^{1,p}( \mathbb{R}^{N})
\end{equation}
has an unbounded sequence of symmetric solutions.
\end{theorem}

Next we give an example where the hypotheses of Theorem
\ref{thm4} are satisfied.

\begin{example}\label{examp1} \rm
Let $2\leq N<p<+\infty$ and $h:\mathbb{R}\to\mathbb{R}$ be defined by
$$
h(z)=\begin{cases}
e^z, & z<2,\\
0,& z=2,\\
z^{2}, & z>2.
\end{cases}
$$ 
The function $h$ has only one discontinuity point at $z_{0}=2$ 
where $h(z_{0})=0$.
Hence, the condition (A9) is satisfied.
 A direct calculation shows that
$$
H(z)=\begin{cases}
e^z-1, & z<2,\\
0,& z=2,\\
z^{3}/3, & z>2.
\end{cases}
$$ 
Therefore,
$$
\liminf_{\xi\to +\infty}\frac{\sup_{|t|\leq \xi}H(t)}{\xi^{p}}=0, \quad
\limsup_{\xi\to+\infty}\frac{H(\xi)}{\xi^p}=+\infty,
$$ 
and we observe that (A8) is fulfilled. Hence, using Theorem \ref{thm4},
the problem
$$
-\Delta_{p}u+| u |^{p-2}u
=\lambda \frac{h(u)}{(1+|x|^2)^2}+\mu\frac{ g_1(u)}{1+|x|^2}, \quad 
 x\in \mathbb{R}^{N},
$$
where
$$
g_1(z)=\begin{cases}
e^z, & z<2,\\
0,& z\geq2.
\end{cases}
$$
for every $(\lambda,\mu)\in]0,+\infty[\times[0,+\infty[$ admits an unbounded
sequence of radially symmetric solutions in
$W_{r}^{1,p}(\mathbb{R}^N)$.
\end{example}

 Arguing as in the proof of Theorem \ref{thm2}, but using conclusion
(c) of Theorem \ref{thm1} instead of (b), the following result holds.

\begin{theorem}\label{thm5}
Assume that {\rm (A1)} and {\rm(A3)} hold and
\begin{itemize}
\item[(B1)]
\begin{align*}
&\liminf_{\xi\to 0^+}\frac{\int_{0}^{+\infty}(\sup _{|
t|\leq\xi}
 F(\rho,t))\rho^{N-1}d\rho}{\xi^{p}}\\
&< \frac{1}{k^p\omega_{N}
D^{N}(\frac{2^p}{D^p}(1-\frac{1}{2^N})+1)}\limsup_{\xi\to
0^+} \frac{\int_{0}^{\frac{D}{2}}F(\rho,\xi)\rho^{N-1}d\rho
}{\xi^p} .
\end{align*}
\end{itemize}
Put
\begin{gather*}
\lambda_3:=\frac{\omega_{N}
D^{N}(\frac{2^p}{D^p}(1-\frac{1}{2^N})+1)}{p\limsup_{\xi\to
0^+}\frac{\int_{0}^{\frac{D}{2}}F(\rho,\xi)\rho^{N-1}d\rho
}{\xi^p}},\\
\lambda_4:=\frac{1}{pk^p\liminf_{\xi\to
0^+}\frac{\int_{0}^{+\infty}(\sup_{| t|\leq\xi}
 F(\rho,t))\rho^{N-1}d\rho}{\xi^{p}}}.
\end{gather*}
Suppose that $g:\mathbb{R}^{N}\times\mathbb{R}\to
\mathbb{R}$ is continuous  almost everywhere, and that
for $\delta_2 >0 $ there is a constant $M_{\delta_2}$ such that
\eqref{6} holds, and satisfies (A4), whose potential
$G(\rho,t)=\int^{t}_{0}g(\rho,s)ds$, $\rho\in \mathbb{R}^{+}
\cup{\{0\}}$, $t \in\mathbb{R}$ is a non-negative
function satisfying the condition
\begin{equation}\label{19} 
g_0:=\lim_{\xi\to 0^+}\frac{\int_{0}^{+\infty}(\sup_{|
t|\leq\xi}G(\rho,t))
\rho^{N-1}d\rho}{\xi^p}<+\infty
\end{equation} 
and set
$$
\bar{\mu}_{g,\lambda}:=\frac{1}{pk^{p} g_{0}}
\Big(1-\lambda p{k}^p\liminf_{\xi\to
0^+}\frac{\int_{0}^{+\infty}(\sup_{|
t|\leq\xi}F(\rho,t))\rho^{N-1}d\rho}{\xi^{p}}\Big).
$$
Then for each $\lambda\in]\lambda_3, \lambda_4[$ and for every
$\mu\in[0,\bar{\mu}_{g,\lambda}[$,  problem \eqref{1} has a
sequence of symmetric solutions, which strongly converges to $0$
in $W_{r}^{1,p}(\mathbb{R}^{N})$.
\end{theorem}

\begin{proof}
We take $X$, $\Phi$, $\Upsilon$, $j$, $\Psi$ and $I_\lambda$ as in the proof 
of Theorem \ref{thm2}. By a similar way as in
the proof of Theorem \ref{thm2} we show that $\delta<+\infty$. For
this, let $\{\xi_n\}$ be a sequence of positive numbers such that
$\xi_n\to0^{+}$ as $n\to+\infty$ and
$$
\lim_{n\to+\infty}\frac{\int_{0}^{+\infty}
(\sup_{|t|\leq\xi_{n}}[F(\rho,t)
+\frac{\mu}{\lambda}G(\rho,t)])\rho^{N-1}d\rho}{\xi_n^{p}}
<+\infty.
$$
Setting $r_n=\frac{1}{p}(\frac{ \xi_n}{k})^{p}$ for all
$n\in \mathbb{N}$, arguing as in the proof of Theorem \ref{thm2}, it
follows that $\delta<+\infty$. 
Fix $\lambda\in ]\lambda_3, \lambda_4[$. The functional $I_\lambda$ 
does not have a local minimum at zero. Indeed, let $\{d_n\}$ be a 
sequence of positive numbers and $\tau>0$ such that $d_n\to 0^{+}$ as
 $n\to \infty$ and
\begin{equation}\label{20}
\frac{1}{\lambda}<\tau<\frac{p}{\omega_{N}D^{N}(\frac{2^p}
{D^p}(1-\frac{1}{2^N})+1)}
\frac{\int_{0}^{\frac{D}{2}}[F(\rho,d_n)
+\frac{\overline{\mu}}{\overline{\lambda}}G(\rho,d_n)]
\rho^{N-1}d\rho}{d_{n}}
\end{equation} 
for each $n\in\mathbb{N}$ large enough. Let $\{w_n\}$ be a sequence in 
$W_{r}^{1,p}(\mathbb{R}^{N})$ defined as given in \eqref{14}.
According to \eqref{15}, \eqref{16} and \eqref{20}, we obtain
\begin{align*}
I_{\lambda}(w_n)
&\leq \frac{1}{p} d_{n}^p \omega_{N} D^N\big(\frac{2^p}{D^p}(1-\frac{1}{2^N})+1\big)
 -\lambda \int_{0}^{\frac{D}{2}}[F(\rho,d_n)+\frac{\mu}{\lambda}G(\rho,d_n)]
\rho^{N-1}d\rho\\
&<\frac{1}{p}d_{n}^p \omega_{N} D^N\big(\frac{2^p}{D^p}(1-\frac{1}{2^N})+1\big)
(1-{\lambda}\tau)<0
\end{align*}
for every $n\in \mathbb{N}$ large enough. Since $I_\lambda(0)=0$, this
means the functional $I_\lambda$ does not have a local minimum at
zero. Hence, the part (c) of Theorem \ref{thm1} concludes that there
exists a sequence $\{u_n\}$ in $X$ of critical points of
$I_\lambda$ such that $\|u_n\|_{ W_{r}^{1,p}(\mathbb{R}^{N})}\to
0$ as $n\to \infty$, and the proof is complete.
\end{proof}

\begin{remark}\label{rmk3}\rm
Note that Assumption (B1) in Theorem
\ref{thm5} could be replaced by the more general condition
\begin{itemize}
\item[(B2)] there exist two sequences $\{\alpha_n\}$ and $\{\beta_n\}$
with 
\[
\Big(\omega_{N}D^N(\frac{2^p}{D^p}
 (1-\frac{1}{2^N})+1)\Big)^{1/p}\alpha_n 
<\frac{\beta_n}{k}
\]
 for every $n\in \mathbb{N}$ and 
$\lim_{n\to+\infty}\beta_n=0$ such that
\begin{align*}
&\lim_{n\to +\infty}\frac{\int_{0}^{+\infty}(\sup_{| t|\leq\beta_n}
F(\rho,t))\rho^{N-1} d\rho-
\int_{0}^{\frac{D}{2}}F(\rho,\alpha_{n})\rho^{N-1}d\rho}{({{\frac{
\beta_n}{k}})^{p}
-\omega_{N}D^N(\frac{2^p}{D^p}(1-\frac{1}{2^N})+1)}
\alpha_{n}^p}\\
&<\frac{1}{\omega_{N} D^N (\frac{2^p}{D^p}
(1-\frac{1}{2^N})+1)}\limsup_{\xi\to
0^+}\frac{\int_{0}^{\frac{D}{2}}F(\rho,\xi)\rho^{N-1}d\rho}{
\xi^p}.
\end{align*}
\end{itemize}
\end{remark}

\begin{remark}\label{rmk4}\rm
We  observe that in Theorem
\ref{thm3}, Corollary \ref{c1} and Theorem \ref{thm4}
 by Theorem \ref{thm5} and replacing $\xi\to+\infty$ with $\xi\to 0^+$, by the same
reasoning, we have the conclusions, $\xi\to+\infty$ replaced by
$\xi\to 0^+$, but the sequences of symmetric solutions strongly
converge to $0$ in $W_{r}^{1,p}(\mathbb{R}^{N})$,
instead.
\end{remark}

 We here give the following example to illustrate our results.

 \begin{example}\label{examp2}
Put $N=2$ and $p=3$. Let $f:\mathbb{R}\to\mathbb{R}$ be defined by
$$
f(z)=\begin{cases}
1, & (z-1)\in[0,1]\setminus C,\\
0, & \text{otherwise}
\end{cases}
$$
 where $C$ is the
``middle third set'' of Cantor. 
Clearly, $m(D_f) = m(1+C) = 0$ and
for each $z\in D_f$ one has $f(z)=0$. A direct calculation shows
$$
F(z)=\begin{cases}
z, & (z-1)\in[0,1]\setminus C,\\
0,& \text{otherwise}.
\end{cases}
$$ 
Therefore,
\[
\liminf_{\xi\to0^+}\frac{\sup_{|t|\leq \xi}F(t)}{\xi^{3}}=0, \quad 
\limsup_{\xi\to0^+}\frac{F(\xi)}{\xi^3}=+\infty.
\]
Hence, taking Remark \ref{rmk4} into account, by the similar result to Theorem
\ref{thm5}, for a fixed continuous almost everywhere function
$g:\mathbb{R}^N\times\mathbb{R}\to \mathbb{R}$ satisfying the
required assumptions in Theorem \ref{thm5}, the problem
\[
-\Delta_{3}u+| u | u=\lambda f(u)+\mu g(|x|,u), \quad  x\in \mathbb{R}^{2},
\quad u\in W_{r}^{1,3}(\mathbb{R}^{2}),
\]
 for every $
\lambda\in]0,+\infty[$ and $\mu$ lying in a convenient interval,
admits a sequence of symmetric solutions, which converges strongly 
to $0$ in $W_{r}^{1,3}(\mathbb{R}^{2})$.
\end{example}

 We now consider the problem
\begin{equation}\label{21}
-\Delta_{p}u+| u |^{p-2}u=\lambda \alpha(x)f(u)+\mu \beta(x)g(u), 
\quad  x\in \mathbb{R}^{N},\quad
u \in W^{1,p}(\mathbb{R}^{N})
\end{equation} 
where $\lambda >0$ and $\mu\geq0$ are two parameters, 
$\alpha,\beta\in L^1(\mathbb{R}^{N})\cap L^\infty(\mathbb{R}^{N})$
are radially symmetric, $\alpha,\beta\geq 0$,
$\alpha,\beta\not\equiv 0$, $f, g:\mathbb{R}\to
\mathbb{R}$ are  non-negative continuous almost everywhere,
namely, $m(D_{f})=0$ where
$D_{f}= \{z\in \mathbb{R}: f(z)  \text{ is discontinuous at }  z\}$,
and $m(D_{g})=0$ where 
$D_{g}= \{ z\in \mathbb{R}, g(z)  \text{ is discontinuous at } z\}$.
We also assume that for each $ \iota_1>0 $ there is a
constant $M_{\iota_1}$ such that
\begin{equation}\label{22}
\sup_{|z|\leq \iota_1}| f( z )|\leq M_{\iota_1}. 
\end{equation}
Let $k_{\infty}$ be the embedding constant of 
$W^{1,p}(\mathbb{R}^{N}) \subset L^{\infty}(\mathbb{R}^{N}) $; we
obtain
$$
\sup_{x\in\mathbb{R}^N}|u(x)|\leq k_{\infty}\|u\|_{W^{1,p}(\mathbb{R}^{N})},
$$ 
and  $k_{\infty}\leq 2p(p-N)^{-1}$ (see \cite{K1}). Put
$$
F(t)=\int^{t}_{0}f(s)ds,\quad t \in\mathbb{R}.
$$
 
Next we have an existence result under the following assumptions:


\begin{itemize}
\item[(A10)] 
\[
\liminf _{\xi\to+\infty}
\frac{\|\alpha\|_{L^1(\mathbb{R}^N)}F(\xi)}{\xi^{p}} <
\frac{\|\alpha\|_{L^1(S(0,\frac{D}{2}))}}{k_{\infty}^p \omega_{N}
D^{N}(\frac{2^p}{D^p}(1-\frac{1}{2^N})+1)}
\limsup_{\xi\to+\infty}\frac{F(\xi)}{\xi^p};
\]

\item[(A11)] for all $ z\in D(f) $ the condition 
$ f^{-} (z) \leq 0 \leq f^{+}(z) $ implies $ f(z)=0 $,
where
\[
 f^{-}(z)={\lim_{\delta\to
0^{+}}}\operatorname{ess\,inf}_{| z-\zeta | <\delta}f(\zeta),
f^{+}(z)={\lim_{\delta\to 0^{+}}}\operatorname{ess\,sup}_{| z-\zeta |
 <\delta}f(\zeta).
\]
\end{itemize}
Put
\begin{gather*}
\lambda_5 : =\frac{\omega_{N} D^{N}(\frac{2^p}{D^p}
(1-\frac{1}{2^N})+1)}{p\|\alpha\|_{L^1(S(0,\frac{D}{2}))}
\limsup_{\xi\to+\infty}\frac{F(\xi)}{\xi^p}},
\\
\lambda_6:=\frac{1}{pk_\infty\|\alpha\|_{L^1(\mathbb{R}^N)}
\liminf _{\xi\to+\infty} \frac{F(\xi) }{\xi^{p}}}.
\end{gather*}
Suppose that $g:\mathbb{R}\to \mathbb{R}$ is a non-negative continuous 
almost everywhere function such that for each $\iota_{2} >0 $ there is a constant
$M_{\iota_{2}}$ such that
\begin{equation}\label{23}
\sup_{| z|\leq \iota_{2}}g( z)\leq M_{\iota_{2}},
\end{equation}
\begin{itemize}
\item[(A12)]
 for all $ z\in D(g) $ the condition $g^{-} (z) \leq 0 \leq g^{+}(z) $ 
implies $g(z)=0 $,
where $g^{-}(z)=\lim_{\delta\to 0^{+}} \operatorname{ess\,inf}_{|
z-\zeta | <\delta}g(\zeta), g^{+}(z)=\lim_{\delta\to
0^{+}} \operatorname{ess\,sup}_{| z-\zeta | <\delta}g(\zeta)$, whose
potential $G(t)=\int^{t}_{0}g(s)ds, \ t \in\mathbb{R}$, is a
non-negative function satisfying the condition
\begin{equation}\label{24} 
g'_\infty:=\|\beta\|_{L^1(\mathbb{R}^N)}\lim_{\xi\to+\infty}
\frac{G(\xi)}{\xi^p}<+\infty
\end{equation} 
and set
$$
\bar{\mu}'_{g,\lambda}:=\frac{1}{pk_{\infty}^{p} g'_{\infty}}
\Big(1-\lambda pk_{\infty}^p \|\beta\|_{L^1(\mathbb{R}^N)} \liminf_{\xi\to
+\infty}\frac{F(\xi)}{\xi^{p}}\Big).
$$ 
\end{itemize}


\begin{theorem}\label{thm6}
Under assumptions {\rm (A10)--(A12)}, for each
$\lambda\in]\lambda_5, \lambda_6[$ and for every
$\mu\in[0,\bar{\mu}'_{g,\lambda}[$, problem \eqref{21} has an
unbounded sequence of symmetric solutions in
$W_{r}^{1,p}(\mathbb{R}^{N})$.
\end{theorem}


We remark that no symmetry requirements on the nonlinear terms $f$ and $g$
are needed.

\begin{proof}[Proof of Theorem \ref{thm6}]
Fix $\lambda$ and $\mu$ as in the conclusion. 
Take $X=W^{1,p}(\mathbb{R}^{N})$ and define the
functionals
\begin{gather*}
\Phi(u)=\frac{1}{p}\| u  \|^{p}_{W^{1,p}(\mathbb{R}^{N})},\quad
 \Upsilon(u)=\int_{\mathbb{R}^{N}}[\alpha(x)F(u(x))
+\frac{\mu}{\lambda}\beta(x)G(u(x))]dx,\\
j(u)=0, \quad 
\Psi(u)=\Upsilon(u)-j(u)=\Upsilon(u)
\end{gather*}
for each $u\in X$. Put $I_\lambda=\Phi-\lambda\Psi$.
 Clearly, the functional $\Phi$ is locally Lipschitz and
weakly sequentially lower semi-continuous. Since $f$ and $g$
satisfy \eqref{22} and \eqref{23}, respectively, and
$W^{1,p}_r(\mathbb{R}^N )$ is compactly embedded in
$L^\infty(\mathbb{R}^N)$, the assertion remains true regarding
$I_{\lambda}$ too. Moreover, like for Theorem \ref{thm2}, we obtain
that any critical point $u \in W^{1,p}(\mathbb{R}^N )$ of the
functional $I_{\lambda}$ is a solution of the problem \eqref{21}.
Thanks to a non-smooth version of the principle of symmetric
criticality introduced by Krawcewicz and Marzantowicz \cite{KM},
we can obtain any critical point of $I_\lambda^r =
I_\lambda|_{W^{1,p}_ r(\mathbb{R}^N )}$ will be also a critical
point of $I_\lambda$. Consider a real sequence $\{d_n\}$ such that
$d_n\to +\infty$ as $n\to \infty$. Let $\{w_n\}$ be a sequence in
$W^{1,p}(\mathbb{R}^N )$ defined as in \eqref{14}. It is easy to
verify that $w_n \in W^{1,p}(\mathbb{R}^N )$ and it is radially
symmetric. Since $0\leq w_n(x)\leq d_n$ for every
$x\in\mathbb{R}^N$, and $f$ and $\alpha$ are non-negative, one has
$$
\int_{S(0,D)\setminus S(0,\frac{D}{2})}\alpha(x)F(w_n(x))dx\geq 0.
$$
Hence, one has
\begin{align*}
\int_{\mathbb{R}^N }\alpha(x)F(w_n(x))dx
&=\int_{ S(0,\frac{D}{2})}\alpha(x)F(w_n(x))dx
 +\int_{S(0,D)\setminus S(0,\frac{D}{2})}\alpha(x)F(w_n(x))dx\\
&\geq \int_{ S(0,\frac{D}{2})}\alpha(x)F(w_n(x))dx\\
&=\omega_N(\frac{D}{2})^{N}\|\alpha\|_{L^1(S(0,\frac{D}{2}))}F(d_n).
\end{align*}
Then, from (A10) we have
\begin{align*}
\liminf _{\xi\to+\infty}
\frac{\|\alpha\|_{L^1(\mathbb{R}^N)}F(\xi)}{\xi^{p}} 
&< \frac{\|\alpha\|_{L^1(S(0,\frac{D}{2}))}}{k_{\infty}^p \omega_{N}
D^{N}(\frac{2^p}{D^p}(1-\frac{1}{2^N})+1)}
\limsup_{\xi\to+\infty}\frac{F(\xi)}{\xi^p}\\
&\leq \frac{\int_{\mathbb{R}^N }\alpha(x)dx}{k_{\infty}^p \omega_{N}
D^{N}(\frac{2^p}{D^p}(1-\frac{1}{2^N})+1)}
\limsup_{\xi\to+\infty}\frac{F(\xi)}{\xi^p}.
\end{align*}
As in Theorem \ref{thm2}, we can prove that the functional
$I_{\overline{\lambda}}$ is unbounded from below, and it follows
that $I_{\overline{\lambda}}$ has no global minimum. Therefore,
applying Theorem \ref{thm1} we deduce that there is a sequence
$\{u_n\}\subset W_{r}^{1,p}(\mathbb{R}^N )$ of critical points of
$I_{\overline{\lambda}}$ such that $\lim_{n\to\infty}\| u_{n}\|_{
W_{r}^{1,p}(\mathbb{R}^{N})}=+\infty$. Hence, we have the
conclusion.
\end{proof}

\begin{remark}\label{rmk5} \rm
We also observe that in Theorem \ref{thm6} by Theorem \ref{thm5} 
and replacing $\xi\to+\infty$ with
$\xi\to 0^+$, by the same reasoning, we have the conclusions,
$\xi\to+\infty$ replaced by $\xi\to 0^+$, but the sequences of
symmetric solutions strongly converge to $0$ in
$W_{r}^{1,p}(\mathbb{R}^{N})$, instead.
\end{remark}

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\end{document}