\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2017 (2017), No. 251, pp. 1--10.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2017 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2017/251\hfil Existence of solutions] {Existence of solutions to superlinear p-Laplace equations without Ambrosetti-Rabinowizt condition} \author[D. M. Duc \hfil EJDE-2017/251\hfilneg] {Duong Minh Duc} \address{Duong Minh Duc \newline University of Sciences, Vietnam National University, 227 Nguyen Van Cu Q5, Hochiminh City, Vietnam} \email{dmduc@hcmus.edu.vn} \thanks{Submitted May 8, 2017. Published October 10, 2017.} \subjclass[2010]{46E35, 35J20} \keywords{Nemytskii operators; p-Laplacian; multiplicity of solutions; \hfill\break\indent mountain-pass theorem} \begin{abstract} We study the existence of non-trivial weak solutions in $W^{1,p}_{0}(\Omega)$ of the super-linear Dirichlet problem \begin{gather*} - \operatorname{div}(|\nabla u|^{p-2}\nabla u)=f(x,u) \quad \text{in }\Omega,\\ u=0 \quad \text{on }\partial\Omega, \end{gather*} where $f$ satisfies the condition \[ |f(x,t)|\leq |\omega(x)t|^{r-1} + b(x)\quad \forall (x,t) \in \Omega\times \mathbb{R}, \] where $r\in (p,\frac{Np}{N-p})$, $b\in L^{\frac{r}{r-1}}(\Omega)$ and $|\omega|^{r-1}$ may be non-integrable on $\Omega$. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{example}[theorem]{Example} \allowdisplaybreaks \section{Introduction} Let $N$ be an integer $\ge 3$, $\Omega$ be a bounded domain in $\mathbb{R}^N$ with smooth boundary $\partial{\Omega}$, $p$ be in $[1,N)$ and $p^{\ast}=\frac{Np}{N-p}$. Let $W^{1,p}_{0}(\Omega)$ be the usual Sobolev space with the following norm \[ \|u\|_{1,p} = \Big\{\int_{\Omega}|\nabla u|^{p}dx\Big\}^{1/p}\quad \forall u\in W^{1,p}_{0}(\Omega). \] We consider the Dirichlet problem \begin{equation} \begin{gathered} - \operatorname{div}(|\nabla u|^{p-2}\nabla u)=f(x,u) \quad \text{in }\Omega,\\ u=0 \quad \text{on }\partial\Omega, \end{gathered} \label{eqP} \end{equation} %\label{th00} where $f$ is a real Carath\'{e}odory function on $\Omega \times \mathbb{R}$ and satisfies the following conditions \begin{itemize} \item[(A1)] there exist $r\in (p,p^{\ast})$, $\omega \in \mathcal{K}_{p,r}$ (see Definition \ref{def0}) and $b\in L^{\frac{r}{r-1}}(\Omega)$ such that \[ |f(x,t)|\leq |\omega(x)t|^{r-1} + b(x)\quad \forall (x,t) \in \Omega\times \mathbb{R}, \] \item[(A2)] there exist $C\in [0,\infty)$ and $d\in L^{1}(\Omega)$ such that $|f(x,t)|\leq d(x)$ for every $x$ in $\Omega$ and $|t|\le C$, \item[(A3)] there is $d_1$ in $L^{\frac{N}{p}}(\Omega)$ such that $ d_1(x)\le \frac{f(x,t)}{|t|^{p-2}t}$ for every $(x,t)\in \Omega\times\mathbb{R}$, \item[(A4)] $f(x,0)=0$ for every $x$ in $\Omega$ and $ \lim_{t\to 0}\frac{f(x,t)}{|t|^{p-2}t} =0$ a.e. in $\Omega$, and \item[(A5)] $ \lim_{|t|\to \infty} \frac{f(x,t)}{|t|^{p-2}t} =\infty$ a.e. in $\Omega$. \end{itemize} The integrability of $|\omega|^{r-1}$ is essential in \cite{AR,DJM,DV,LW,LWZ,LIUc,LIUb,MS,NM}, because these papers have used the differentiability of Nemytskii from $L^{q_1}(\Omega)$ into $L^{q_{2}}(\Omega)$ (see \cite{DJM,FI}) and the Sobolev embedding from $W_{0}^{p}(\Omega)$ into $L^{q_1}(\Omega)$. In the present paper, using weighted Sobolev embeddings in \cite{DUC,KJF,KU,MA,OK} instead of classical one in \cite{BR}, we can study the problem \eqref{eqP} with non-integrable functions $|\omega|^{r-1}$ in (A1). In many applications, $ \frac{f(x,t)}{|t|^{p-2}t}$ is non-negative for $t\neq 0$ and $|f(x,t)|$ is well-controlled when $|t|$ is sufficiently small. This observation is the motivation of (A2) and (A3). Here we consider the case, in which the positivity of $ \frac{f(x,t)}{|t|^{p-2}t}$ can be disturbed by a function $d_1\in L^{\frac{N}{p}}(\Omega)$. In (A4) and (A5), we do not need the uniform convergence as in \cite{AR,DJM,DV,LW,LWZ,LIUc,LIUb,MS,NM}. We study the problem \eqref{eqP} without Ambrosetti-Rabinowizt condition. Our main result is the following theorems under the assumption \begin{itemize} \item[(A6)] $\frac{f(x,t)}{|t|^{p-2}t}$ is increasing in $t \ge C$ and decreasing in $t \le -C$ for every $x$ in $\Omega$. \end{itemize} \begin{theorem} \label{theorem2} Assume $f$ satisfies {\rm (A1)-(A6)}. Then there is a non-trivial weak solution in $W^{1,p}_{0}(\Omega)$ of the problem \eqref{eqP}. \end{theorem} \begin{remark} \rm If $f$ is continuous on $\overline{\Omega}\times\mathbb{R}$ and satisfies the following conditions \begin{itemize} \item[(A1')] There exist $r\in (p,p^{\ast}-1)$ and a positive real number $\alpha$ such that \[ |f(x,t)|\leq \alpha(1+|t|^{r-1})\quad \forall (x,t) \in \Omega\times \mathbb{R}. \] \item[(A4')] $f(x,0)=0$ for every $x$ in $\Omega$ and $ \lim_{t\to 0}\frac{f(x,t)}{|t|^{p-2}t} =0$ uniformly in $\Omega$. \item[(A5')] $\lim_{|t|\to \infty}\frac{f(x,t)}{|t|^{p-2}t} =\infty$ uniformly in $\Omega$. \end{itemize} Then $f$ satisfies (A1)--(A5). Therefore our theorem improves the corresponding results in \cite{LIUb,MS}. \end{remark} We study a method for constructing weight functions in weighted Sobolev embeddings and the Nemytskii operator from Sobolev spaces into Lebesgue spaces (see Theorems \ref{th2} and \ref{th3}). We apply these results to prove the existence of non-trivial solutions of a class of super-linear p-Laplacian problem in the last section. \section{Nemytskii operators} \begin{definition} \label{def0}\rm Let $\sigma$ be a measurable function on $\Omega$. We put \[ T_{\sigma}u = \sigma u\quad h\forall u\in W^{1,p}_{0}(\Omega). \] We say that \begin{itemize} \item[(i)] $\sigma$ is of class $\mathcal{C}_{p,s}$, if $T_{\sigma}$ is a continuous mapping from $W^{1,p}_{0}(\Omega)$ into $L^{s}(\Omega)$, \item[(ii)] $\sigma$ is of class $\mathcal{K}_{p,s}$, if $T_{\sigma}$ is a compact mapping from $W^{1,p}_{0}(\Omega)$ into $L^{s}(\Omega)$. \end{itemize} \end{definition} We have following results. \begin{theorem} \label{th01} Let $\alpha_1$ and $\alpha_{2}$ be in $[1,\infty)$ such that $\alpha_1 <\alpha_{2}$. Let $\omega_1\in \mathcal{C}_{p,\alpha_1}$, $\omega_{2}\in \mathcal{C}_{p,\alpha_{2}}$ such that $\omega_1$ and $\omega_{2}$ are non-negative. Let $\beta\in (\alpha_1, \alpha_{2})$ and $\omega = \omega_1^{\frac{\alpha_1(\alpha_{2}-\beta)}{\beta(\alpha_{2} -\alpha_1)}}\omega_{2}^{\frac{\alpha_{2}(\beta-\alpha_1)}{\beta(\alpha_{2} -\alpha_1)}}$. Then $w \in \mathcal{C}_{p,\beta}$. \end{theorem} \begin{proof} There is a positive real number $C_1$ such that \begin{equation} \Big\{\int_{\Omega}\omega_{i}^{\alpha_{i}}|u|^{\alpha_{i}} dx\Big\}^{1/\alpha_{i}} \le {C_1\lVert u \rVert_{1,p}}\quad h\forall u\in W^{1,p}_{0}(\Omega), \; i= 1,2. \label{oo1} \end{equation} Since $\beta = \frac{\alpha_{2}-\beta}{\alpha_{2} -\alpha_1}\alpha_1 + \frac{\beta-\alpha_1}{\alpha_{2} -\alpha_1}\alpha_{2}$, by H\"{o}lder's inequality and \eqref{oo1}, we get \begin{align*} &\Big\{\int_{\Omega}\omega^{\beta}|u|^{\beta} dx\Big\}^{1/\beta}\\ &=\Big\{\int_{\Omega}\omega_1^{\frac{\alpha_{2}-\beta}{\alpha_{2} -\alpha_1}\alpha_1}|u|^{\frac{\alpha_{2}-\beta}{\alpha_{2} -\alpha_1}\alpha_1}\omega_{2}^{\frac{\beta-\alpha_1}{\alpha_{2} -\alpha_1}\alpha_{2}}|u|^{\frac{\beta-\alpha_1}{\alpha_{2} -\alpha_1}\alpha_{2}} dx\Big\}^{1/\beta} \\ &\le \Big\{\Big\{\int_{\Omega}\omega_1^{\alpha_1}|u|^{\alpha_1} dx\Big\} ^{\frac{\alpha_{2}-\beta}{\alpha_{2} -\alpha_{2}}} \Big\{\int_{\Omega}\omega_{2}^{\alpha_{2}}|u|^{\alpha_{2}} dx \Big\}^{\frac{\beta-\alpha_1}{\alpha_{2} -\alpha_1}}\Big\}^{1/\beta} \\ &\le \Big\{\Big\{\int_{\Omega}\omega_1^{\alpha_1}|u|^{\alpha_1} dx \Big\}^{\frac{1}{\alpha_1}\frac{\alpha_{2}-\beta}{\alpha_{2} -\alpha_{2}}\alpha_1}\Big\{\int_{\Omega}\omega_{2}^{\alpha_{2}}|u|^{\alpha_{2}} dx \Big\}^{\frac{1}{\alpha_{2}}\frac{\beta-\alpha_1}{\alpha_{2} -\alpha_1}\alpha_{2}} \Big\}^{1/\beta}\\ &\le {C_1\lVert u \rVert_{1,p}}\quad \forall u \in W^{1,p}_{0}(\Omega). \end{align*} \end{proof} \begin{theorem} \label{th1} Let $s$ be in $[1,\frac{Np}{N-p})$, $\alpha$ be in $(0,1)$, $\omega \in \mathcal{C}_{p,s}$ and $\theta$ be measurable functions on $\Omega$ such that $\omega \ge 0$ and $|\theta|\le\omega^{\alpha}$. Then $\theta$ is of class $\mathcal{K}_{p,s}$. \end{theorem} \begin{proof} Since $T_{\omega}$ is in $\mathcal{C}_{p,s}$, $T_{\omega}$ is continuous from $W^{1,p}_{0}(\Omega)$ into $L^{s}(\Omega)$ and there is a positive real number $C_{2}$ such that \begin{equation} \Big\{\int_{\Omega}|u|^{s}\omega^{s} dx\Big\}^{1/s} \le C_{2}\lVert u \rVert_{1,p}\quad \forall u \in W^{1,p}_{0}(\Omega).\label{o1} \end{equation} Since $\omega^{\alpha}(x) \le 1 +\omega(x)$ for every $x$ in $\Omega$ and $1$ and $\omega$ are in $\mathcal{C}_{p,s}$, $\omega^{\alpha}$ belongs to $\mathcal{C}_{p,s}$. Thus $ T_{\theta} $ is in $\mathcal{C}_{p,s}$. Let $M$ be a positive real number and $\{u_{n}\}$ be a sequence in $W^{1,p}_{0}(\Omega)$, such that $\lVert u_{n}\rVert_{1,p}\le M$ for any $n$. By Rellich-Kondrachov's theorem \cite[Theorem 9.16]{BR}, $\{u_{n}\}$ has a subsequence $\{u_{n_{k}}\}$ converging to $u$ in $L^{s}(\Omega)$ and $ \{u_{n_{k}}\}$ converging weakly to $ u $ in $ W_0^{1,p}(\Omega) $, therefore $ \|u\|_{1,p} \leq \liminf_{k \to \infty} \|u_{n_{k}}\|_{1,p} \leq M$. We shall prove $\{T_{\theta}(u_{n_{k}})\}$ converges to $T_{\theta}(u)$ in $L^{s}(\Omega)$. Let $\varepsilon$ be a positive real number. Choose a positive real number $\delta$ such that \begin{equation} (2C_{2}M)^{s}\delta^{(\alpha-1)s} < \frac{\varepsilon^{s}}{2}. \label{th1.1} \end{equation} Put $\Omega' =\{x\in \Omega : \omega(x)> \delta\}$. By \eqref{o1} and \eqref{th1.1}, we have \begin{align} &\int_{\Omega}|\theta(u_{n_k}-u)|^{s}dx \nonumber \\ &= \int_{\Omega}|u_{n_k}-u|^{s}|\theta|^{s}dx \nonumber\\ &\leq \int_{\Omega'}|u_{n_k}-u|^{s}\omega^{\alpha s}dx +\int_{\Omega\setminus \Omega'}|u_{n_k}-u|^{s}\omega^{\alpha s}dx \nonumber\\ &\leq \delta^{(\alpha-1)s}\int_{\Omega'}|u_{n_k}-u|^{s}\omega^{s} dx +\delta^{\alpha s}\int_{\Omega\setminus \Omega'}|u_{n_k}-u|^{s}dx \nonumber\\ &\leq\delta^{(\alpha-1)s}\int_{\Omega}|u_{n_k}-u|^{s}\omega^{s} dx + {\delta^{\alpha s}}\int_{\Omega}|u_{n_k}-u|^{s}dx \nonumber\\ &\leq\delta^{(\alpha-1)s}\left(C_{2}\lVert u_{n_{k}}-u \rVert_{1,p}\right)^{s} + {\delta^{\alpha s}}\int_{\Omega}|u_{n_k}-u|^{s}dx \nonumber\\ &\leq {\delta^{(\alpha-1)s}(2C_{2}M)^{s}} +\delta^{\alpha s} \int_{\Omega}|u_{n_k}-u|^{s}dx \nonumber\\ &\leq \frac{\varepsilon^{s}}{2}+{\delta^{\alpha s}}\int_{\Omega}|u_{n_k}-u|^{s}dx . \label{the1.2} \end{align} Since $\{u_{n_k}\}$ converges in $L^{s}(\Omega)$, there is an integer $k_{0}$ such that \begin{equation} \int_{\Omega}|u_{n_k}-u|^{s}dx \le {\delta^{-\alpha s}}\frac{\varepsilon^{s}}{2} \quad h\forall k\ge k_{0}.\label{the1.3} \end{equation} Combining \eqref{the1.2} and \eqref{the1.3}, we complete the proof. \end{proof} \begin{corollary} \label{co0th0} Let $p \in [1,N)$, $s \in (1,\frac{Np}{N-p})$, $\eta \in (\frac{sNp}{Np-s(N-p)},\infty)$ and $\theta \in L^{\eta}(\Omega)$. Then $\theta$ is in $\mathcal{K}_{p,s}$. \end{corollary} \begin{proof} Let $\beta \in (0,1)$ such that $\beta\eta= \frac{sNp}{Np-s(N-p)}$ and $\omega = |\theta|^{1/\beta}$. Then $\omega$ is in $L^{\frac{sNp}{Np-s(N-p)}}(\Omega)$. Since $\frac{Np-s(N-p)}{Np}+\frac{s(N-p)}{Np}=1$, by H\"older's inequality, we have \[ \int_{\Omega}|\omega u|^{s}dx\le \int_{\Omega}(|\omega|^{\frac{sNp}{Np-s(N-p)}}) ^{\frac{Np-s(N-p)}{Np}}\Big(\int_{\Omega}|u|^{\frac{Np}{N-p}}\Big)^{\frac{s(N-p)}{Np}} \quad \forall u\in W^{1,p}_{0}(\Omega), \] which implies that $T_{\omega}$ is continuous at $0$ in $W^{1,p}_{0}(\Omega)$. Thus $T_{\omega}$ is a linear continuous map from $W^{1,p}_{0}(\Omega)$ into $L^{s}(\Omega)$. By Theorem \ref{th1}, is of class $\mathcal{K}_{p,r}$. \end{proof} \begin{example} \label{ex10} \rm Let $N=5$, $p= 3$, $s=4$ and $\Omega =\{x\in \mathbb{R}^{5}: |x| < 1\}$. Then $\frac{sNp}{Np - s(N-p)} =\frac{4.5.3}{5.3-4(5-3)}=\frac{60}{7}<10$. Put $\omega_{0}= |x|^{-\frac{1}{30}}\cos(16|x|)$, then $\omega_{0}$ is in $L^{10}(\Omega)$. Thus by Corollary \ref{co0th0}, $\omega_{0}$ is of class $\mathcal{K}_{p,s}$. \end{example} \begin{corollary} \label{co1th1} Let $p \in [1,N)$, $s \in (1,\frac{Np}{N-p})$, $\alpha$ be in $(0,1)$ and $\eta \in \mathcal{C}_{p,p}$. Then $\theta=\eta^{\alpha\frac{p(p^{\ast}-s)}{s(p^{\ast} -p)}}$ is of class $\mathcal{K}_{p,s}$. \end{corollary} \begin{proof} Put $\omega_1 = \eta$, $\omega_{2} = 1$, $\alpha_1 = p$, $\alpha_{2}= p^{\ast}$, $\beta =s$. By the Embedding theorem of Sobolev, $\omega_{2} \in \mathcal{C}_{p,p^{\ast}}$. By Theorem \ref{th01}, we see that $\eta^{\frac{p(p^{\ast}-s)}{s(p^{\ast} -p)}}\in \mathcal{C}_{p,s}$. Thus by Theorem \ref {th1.1}, $\eta^{\alpha\frac{p(p^{\ast}-s)}{s(p^{\ast} -p)}}$ is of class $\mathcal{K}_{p,s}$. \end{proof} \begin{example} \label{ex11} \rm Let $\Omega =\{x\in \mathbb{R}^{5}: \|x\| < 1\}$, $p=3$, $s = 4$, $\alpha = \frac{3}{4}$ and $\eta(x) = (1-\|x\|^{2})^{-1}$ for every $x$ in $\Omega$. By \cite[Theorem 8.4]{KU}, $\eta \in \mathcal{C}_{p,p}$. Note that $p^{\ast}=\frac{Np}{N-p} = \frac{15}{2}$ and \[ \alpha\frac{p(p^{\ast}-s)}{s(p^{\ast} -p)}= \frac{3}{4}\frac{3}{4}\frac{7}{9}=\frac{7}{16}. \] Put $\theta(x) = (1-\|x\|^{2})^{-\frac{7}{16}}$ for every $x$ in $\Omega$. Then $\theta \in \mathcal{K}_{3,4}$. \end{example} \begin{theorem} \label{th2} Let $s$ be in $(1,p^{\ast})$, $\omega$ be in $\mathcal{K}_{p,s}$, $b$ be in $L^{\frac{s}{s-1}}(\Omega)$ and $g$ be a Caratheodory function from $ \Omega \times \mathbb{R}$ into $\mathbb{R}$. Assume \begin{equation} |g(x,z)| \leq |\omega(x)|^{s-1}|z|^{s-1} + b(x)\quad h \forall (x,z) \in \Omega \times \mathbb{R}. \label{n1} \end{equation} Put $N_{g}(v)(x)= g(x,v(x))$ for $v \in W^{1,p}_{0}(\Omega)$, $x \in \Omega$. We have \begin{itemize} \item[(i)] $N_{g}$ is a continuous mapping from $W_{0}^{1,p}(\Omega)$ into $L^{\frac{s}{s-1}}(\Omega)$. \item[(ii)] If $A$ is a bounded subset in $W_{0}^{1,p}(\Omega)$, then $\overline{N_{g}(A)}$ is compact in {$L^{\frac{s}{s-1}}(\Omega)$}. \end{itemize} \end{theorem} \begin{proof} (i) Put $\mu= s$, $q= s/(s-1)$ and \[ g_1(x,\zeta)=g(x,\omega(x)^{-1}\zeta)\quad \forall (x,\zeta) \in \Omega \times \mathbb{R}, \] By \eqref{n1}, we have \[ |g_1(x,\zeta)| \leq |\zeta|^{s-1} + b(x)\quad h \forall (x,\zeta) \in \Omega \times \mathbb{R} . \] On the other hand \[ N_{g}(v) = N_{g_1}\circ {T_{|\omega|}(v)}\quad \forall v \in W^{1,p}_{0}(\Omega). \] Since $w \in \mathcal{K}_{p,s}$, applying \cite[Theorem 2.3]{FI}, we complete the proof. \end{proof} \begin{theorem} \label{th3} Let $s\in (1,p^{\ast})$, $\omega$ be in $\mathcal{K}_{p,s}$, a function $b \in L^{\frac{s}{s-1}}(\Omega)$ and $g$ be a Caratheodory function from $ \Omega \times \mathbb{R}$ into $\mathbb{R}$. Assume \[ |g(x,z)| \leq |\omega(x)|^{s-1}|z|^{s-1} + b(x)\quad \forall (x,z) \in \Omega \times \mathbb{R}. \] Put \begin{gather*} G(x,t)=\int_{0}^{t}g(x,\xi)d\xi \quad \forall (x,t) \in \Omega , \\ \Psi_{g}(u)=\int_{\Omega}G(x,t) dx \quad h \forall u \in W^{1,p}_{0}(\Omega) . \end{gather*} We have \begin{itemize} \item[(i)] $\{N_{G}(w_{n})\}$ converges to $N_{G}(w)$ in $L^{1}(\Omega)$ when $\{w_{n}\}$ weakly converges to $w$ in $W^{1,p}_{0}(\Omega)$. \item[(ii)] $\Psi_{g}$ is continuously Fr\'{e}chet differentiable mapping from $W^{1,p}_{0}(\Omega)$ into $\mathbb{R} $ and \[ D\Psi_{g}(u)(\phi)=\int_{\Omega}g(x,\xi)\phi dx \quad h \forall u, \phi \in W^{1,p}_{0}(\Omega) . \] \item[(iii)] If $A$ is a bounded subset in $W^{1,p}_{0}(\Omega)$, then there is a positive real number $M$ such that \[ |\Psi_{g}(v)| + \|D\Psi_{g}(v)\|\le M \quad h\forall v \in {A}. \] \end{itemize} \end{theorem} \begin{proof} Let $\mu=s$, $q= \frac{s}{s-1}$ and $g_1$ be as in the proof of Theorem \ref{th2}. Put \begin{gather*} G_1(x,t) = \int_{0}^{t}g(x,\xi)d\xi \quad \forall (x,t) \in \Omega,\\ \Psi_{g_1}(u)=\int_{\Omega}\int_{0}^{u(x)}g_1(x,\xi)d\xi dx \quad \forall u \in L^{p}(\Omega) . \end{gather*} By \cite[Theorem 2.8]{FI}, $N_{G_1}$ is continuous from $L^{\frac{s}{s-1}}(\Omega)$ to $L^{1}(\Omega)$ and $\Psi_{g_1}$ is continuously Fr\'{e}chet differentiable mapping from $L^{\frac{s}{s-1}}(\Omega)$ to $\mathbb{R} $. We see that $N_{G} = N_{G_1}\circ T_{\omega}$ and $\Psi_{g} = \Psi_{g_1}\circ T_{\omega}$. By Theorem \ref{th1}, we complete the proof. \end{proof} For $\omega=1$, Theorems \ref{th2} and \ref{th3} have been proved in \cite{AZ,FI,KRA}. \begin{example} \label{ex12} \rm Let $\Omega =\{x\in \mathbb{R}^{5}: \|x\| < 1\}$, $p=3$, $s = 4$, $\alpha = \frac{3}{4}$ and $\rho(x) = (\frac{1}{2}-\|x\|^{2})^{2}(1-\|x\|^{2})^{-\frac{7}{16}}$ for every $x$ in $\Omega$. By Example \ref{ex11}, $\rho \in \mathcal{K}_{3,4}$. Put $a(x) = \rho(x)^{s -1} = (\frac{1}{2}-\|x\|^{2})^{6}(1-\|x\|^{2})^{-\frac{21}{16}}$ for every $x$ in $\Omega$. Thus $a$ is not integrable on $\Omega$ and Theorem \ref{th3} improves corresponding results in \cite{AZ,FI,KRA}. \end{example} \section{Proof of main theorems} Put \begin{equation} J(u)=\frac{1}{p}\|u\|_{1,p}^{p}-\int_{\Omega}F(x,u)dx\quad \forall u\in W_{0}^{1,p}(\Omega).\label{eq:2.1} \end{equation} By \cite[Theorem 9]{DJM}, Theorem \ref{th3} and (A1), $J$ is continuously Fr\'{e}chet differentiable on $W_{0}^{1,p}(\Omega)$ and \begin{equation} DJ(u)(v)=\int_{\Omega}|\nabla u|^{p-2}\nabla u.\nabla vdx -\int_{\Omega}f(x,u).vdx~~\forall u,v\in W_{0}^{1,p}(\Omega)\label{DJ} \end{equation} To prove the theorems, we need following lemmas. \begin{lemma} \label{lemma2} Under conditions {\rm (A3)} and {\rm (A4)}, there exists positive numbers $\rho$ and $\eta$ such that $J(u)\geq\eta$ for all $u$ in $W_{0}^{1,p}(\Omega)$ with $\|u\|=\rho$. \end{lemma} \begin{proof} Suppose by contradiction that \[ \inf \{J(u): u \in W^{1,p}_{0}(\Omega), \|u\|_{1,p} = \frac{1}{n}\} \leq 0\quad \forall n\in\mathbb{N}. \] Then there is a sequence $\{u_{n}\}$ in $W^{1,p}_{0}(\Omega)$ such that $\|u_{n}\|_{1,p} =\frac{1}{n}$ and $J(u_{n}) < \frac{1}{n^{p+1}}$. By replacing $\{u_{n}\}$ by its subsequence, by \cite[Theorem 4.9]{BR}, we can suppose that $\lim _{n\to\infty}u_{n}(x) =0$ for every $x$ in $\Omega$, $\{\frac{u_{n}}{\|u_{n}\|_{1,p}}\}$ strongly (resp. pointwisely) converges to $w$ in $L^{p}(\Omega)$ (resp. on $\Omega$) and \begin{align*} \frac{1}{n} &> \frac{J(u_{n})}{\|u_{n}\|_{1,p} ^{p}} \\ &= \frac{1}{p}- \int_{\Omega}\frac{F(x,u_{n}(x))}{\|u_{n}\|_{1,p} ^{p}}dx \\ &= \frac{1}{p}- \int_{\Omega}\int_{0}^{1}f(x,su_{n}(x)) \frac{u_{n}(x)}{\|u_{n}\|_{1,p} ^{p}}\,ds\,dx\\ &=\frac{1}{p}- \int_{\Omega}\int_{0}^{1}\frac{f(x,su_{n}(x))}{(su_{n}(x))^{p-2} su_{n}(x)}s^{p-1}\frac{|u_{n}(x)|^{p}}{\|u_{n}\|_{1,p} ^{p}}\,ds\,dx. \end{align*} Hence by the generalized Fatou Lemma, (A3) and (A4) \begin{align*} 0&= \liminf_{n\to\infty} \frac{1}{n} \\ &= \frac{1}{p} - \limsup_{n\to\infty}\int_{\Omega}\int_{0}^{1} \frac{f(x,su_{n}(x))}{(su_{n}(x))^{p-2}su_{n}(x)}s^{p-1} \frac{|u_{n}(x)|^{p}}{\|u_{n}\|_{1,p} ^{p}}\,ds\,dx\\ &\ge \frac{1}{p} - \int_{\Omega}\int_{0}^{1}\limsup_{n\to\infty} [\frac{f(x,su_{n}(x))}{(su_{n}(x))^{p-2}su_{n}(x)}s^{p-1} \frac{|u_{n}(x)|^{p}}{\|u_{n}\|_{1,p} ^{p}}]\,ds\,dx =\frac{1}{p}. \end{align*} This contradiction completes the proof. \end{proof} \begin{lemma} \label{lemma1} Let $\rho$ be as in Lemma \ref{lemma2}. Under conditions {\rm (A3)} and {\rm (A5)}, there is $e$ in $W^{1,p}_{0}(\Omega)\setminus B(0,\rho)$ such that $J(e)< 0$. \end{lemma} \begin{proof} Let $u\in W^{1,p}_{0}(\Omega)$ such that $\|u\|_{1,p}=1$ and $u >0$ on $\Omega$. By \eqref{eq:2.1}, we have \begin{align*} J(nu) &= \frac{n^{p}}{p}-\int_{\Omega}\int_{0}^{nu(x)}f(x,s)\,ds\,dx \\ &= \frac{n^{p}}{p}-\int_{\Omega}\int_{0}^{1}f(x,\xi nu(x))nu(x)d\xi dx\\ &=\frac{n^{p}}{p}[1-p\int_{\Omega}\int_{0}^{1} \frac{f(x,\xi nu(x))}{(\xi nu(x))^{p-1}}\xi^{p-1} u(x)^{p}d\xi dx]. \end{align*} By Sobolev's embedding theorem, $u$ belongs to $L^{\frac{Np}{N-p}}(\Omega)$. By (A3), $d|u|^{p}$ is integrable and $ \frac{f(x,\xi n u(x))}{|\xi n u(x)|^{p-2}\xi n u(x)}|u(x)|^{p}\ge d(x)|u(x)|^{p}$ for every integer $n$, $x \in \Omega$ and $\xi\in(0,1)$. Hence, by the generalized Fatou lemma, (A3) and (A5), we have \begin{align*} &\limsup_{n\to\infty}[1-p\int_{\Omega}\int_{0}^{1} \frac{f(x,\xi nu(x))}{|\xi n u(x)|^{p-2}\xi n u(x)}\xi^{p-1} |u(x)|^{p}d\xi dx]\\ &=1-\liminf_{n\to\infty}[p\int_{\Omega}\int_{0}^{1} \frac{f(x,\xi nu(x))}{|\xi n u(x)|^{p-2}\xi n u(x)}\xi^{p-1} |u(x)|^{p}d\xi dx]\\ &\le 1- p\int_{\Omega}\int_{0}^{1}\liminf_{n\to\infty} [\frac{f(x,\xi nu(x))}{|\xi n u(x)|^{p-2}\xi n u(x)}\xi^{p-1} |u(x)|^{p}]d\xi dx = -\infty, \end{align*} which implies $\lim_{n\to\infty}J(nu)= -\infty$. \end{proof} \begin{lemma} \label{lemmcc0} Under conditions {\rm (A2)} and {\rm (A6)}, there is a positive real number $C_1$ such that \[ f(x,s)s-pF(x,s)\le f(x,t)t - pF(x,t) +C_1 d(x)\quad \forall x\in\Omega,|s|\le |t|. \] \end{lemma} \begin{proof} By the proof of \cite[Lemma 2.3]{LIUb}, (A2) and (A6), we have \[ f(x,s)s-pF(x,s)\le f(x,t)t - pF(x,t) \quad \forall x\in\Omega,C\le s\le t. \] Let $x\in\Omega$ and $\xi \in [-C,C]$. By (A2), we have \[ |f(x,\xi)\xi|\le Cd(x),\quad |F(x,\xi)| \le \int_{0}^{\xi}d(x) dy\le Cd(x). \] Hence \begin{gather*} f(x,s)s-pF(x,s)\le f(x,t)t - pF(x,t) +2(1+p)Cd(x) \quad \forall x\in\Omega,\; 0\le s\le t\le C, \\ \begin{aligned} f(x,s)s-pF(x,s) &\le f(x,C)C - pF(x,C) +2(1+p)Cd(x)\\ &\le f(x,t)t - pF(x,t) +2(1+p)Cd(x) \quad \forall x\in\Omega,\; 0 \le s\le C \le t. \end{aligned} \end{gather*} Thus we get the lemma when $0\le s\le t$. Similarly we obtain it if $t\le s\le 0$. \end{proof} \begin{lemma} \label{lemmcc1} Assume {\rm (A1)--(A3), (A5), (A6)} hold. Let $\{ u_{n}\}$ be a sequence in $W_{0}^{1,p}(\Omega)$ such that $\{ J(u_{n})\} $ is bounded and $\lim_{n\to \infty}(1+\|u_{n}\|_{1,p})\|DJ(u_{n})\| = 0$. Then $\{ u_{n}\}$ has a subsequence converging in $W_{0}^{1,p}(\Omega)$. \end{lemma} \begin{proof} We shall use the technique in \cite{LIUb,MS}. If $\{ u_{n}\}$ is unbounded, up to a subsequence we may assume that for some $c$ in $\mathbb{R}$ such that $\lim_{n}\|u\|_{1,p}=\infty$, $\lim_{n} J(u)=c$ and $\lim_{n\to \infty}\|u_{n}\|_{1,p}\|DJ(u_{n})\| = 0$. Thus \begin{equation} \begin{aligned} &\lim_{n\to\infty}\int_{\Omega}(\frac{1}{p}f(x,u_{n}(x))u_{n}(x) - F(x,u_{n}(x)))dx\\ & = \lim_{n\to\infty}(J(u_{n})-\frac{1}{p} \langle J'(u_{n}),u_{n}\rangle ) = c. \end{aligned}\label{lemmcc11} \end{equation} Put $w_{n}= \|u_{n}\|_{1,p}^{-1}u_{n}$ for every $n$ in $\mathbb{N}$. Since $\{u_{n}\}$ is bounded in $W^{1,p}_{0}(\Omega)$, by replacing $\{u_{n}\}$ by its subsequence, we can suppose $\{w_{n}\}$ converges weakly to $w$ in $W^{1,p}_{0}(\Omega)$ (resp. strongly in $L^{p}(\Omega)$, pointwisely in $\Omega$). Consider the case $w=0$. By the continuity of $J$, there is $t_{n}$ in $[0,1]$ such that $J(t_{n}u_{n}) = \max\{J(su_{n}) : s \in [0,1]\}$ for every positive integer $n$. Fix a positive integer $m$ and put $v_{n}= (2pm)^{1/p}w_{n}$ for every positive integer $n$. Then $\{v_{n}\}$ converges weakly to $0$ in $W^{1,p}_{0}(\Omega)$ (resp. strongly in $L^{p}(\Omega)$, pointwisely in $\Omega$). Therefore, by Theorem \ref{th3}, $\{N_{F}((v_{n})\}$ converges to $N_{F}(0)=0$ in $L^{1}(\Omega)$. Thus \[ \lim_{n\to\infty}\int_{\Omega}F(x,v_{n}(x))dx =0. \] Since $\lim_{n\to\infty}(2pm)^{1/p}\|u_{n}\|_{1,p}^{-1} =0$, there is an integer $N_{m}$ such that $t_{n}\in [0,1]$ and \[ J(t_{n}u_{n})\ge J(v_{m}) = 2m - \int_{\Omega}F(x,v_{m})\ge m\quad \forall n\ge N_{m}, \] that is, $\lim_{n\to\infty}J(t_{n}u_{n})=\infty$. Since $J(0)=0$ and $\lim_{n\to\infty}J(u_{n})=c$, it implies $t_{n} \in (0,1)$ for any sufficiently large $n$ and \begin{align*} \int_{\Omega}|\nabla (t_{n}u_{n})|^{p} - \int_{\Omega}f(x,t_{n}u_{n})t_{n}u_{n}dx &= \langle J'(t_{n}u_{n}),t_{n}u_{n} \rangle \\ &= t_{n}\frac{d}{dt}|_{t=t_{n}}J(t,u_{n})=0. \end{align*} Therefore, by Lemma \ref{lemmcc0}, we get \begin{align*} &\int_{\Omega}(\frac{1}{p}f(x,u_{n}(x))u_{n}(x)-F(x,u_{n}(x)))dx \\ &\ge \int_{\Omega}(\frac{1}{p}f(x,t_{n}u_{n}(x))t_{n}u_{n}(x)-F(x,t_{n}u_{n}(x)))dx -C_1\|d\|_{L^{1}(\Omega)} \\ &= \int_{\Omega}(\frac{1}{p}|\nabla t_{n}u_{n}(x)|^{p}-F(x,t_{n}u_{n}(x)))dx -C_1\|d\|_{L^{1}(\Omega)} \\ &= J(t_{n}u_{n})-C_1\|d\|_{L^{1}(\Omega)}\to\infty, \end{align*} which contradicts \eqref{lemmcc11}. If $w\neq 0$, the Lebesgue measure of the set $\Theta = \{x\in\Omega : w(x)\neq 0\}$ is positive. We have $\lim_{n\to\infty}|u_{n}(x)|=\infty$ for every $x$ in $\Theta$. Thus, By the generalized Fatou lemma, (A3) and (A5), we have \begin{align*} 0&= \liminf_{n\to\infty}[\frac{1}{p} - \frac{J(u_{n})}{\|u_{n}\|_{1,p}}] = \liminf_{n\to\infty} \int_{\Omega}\frac{F(x,u_{n}(x))}{\|u_{n}\|_{1,p}^{p}}dx \\ &\ge \liminf_{n\to\infty}\Big[\int_{\Theta}\int_{0}^{1} \frac{f(x,\xi u_{n}(x))}{|\xi u_{n}(x)|^{p-2}\xi u_{n}(x)} |\xi w_{n}(x)|^{p} \,d\xi\, dx \\ &\quad +\int_{\Omega\setminus\Theta}\int_{0}^{1}d_1|\xi w_{n}(x)|^{p}d\xi dx\Big]\\ &\ge \int_{\Theta}\int_{0}^{1}\liminf_{n\to\infty} \frac{f(x,\xi u_{n}(x))}{(|\xi u_{n}(x)|^{p-2}\xi u_{n}(x)}|\xi w_{n}(x)|^{p}\,d\xi\,dx \\ &\quad - \|d_1\|_{L^{\frac{n}{p}}(\Omega)}\|w_{n}\|_{L^{\frac{Np}{N-p}}(\Omega)} =\infty, \end{align*} which is impossible. In any case, we obtain a contradiction. Therefore $\{u_{n}\}$ is bounded. By Theorem \ref{th2}, there is a subsequence $\{u_{n_{k}}\}$ of $\{u_{n}\}$ such that $\{u_{n_{k}}\}$ weakly converges to $u$ in $W^{1,p}_{0}(\Omega)$ and $\{N_{f}(u_{n_{k}})\}$ weakly converges to $N_{f}(u)$ in $L^{\frac{p}{p-1}}(\Omega)$. Arguing as in the proof of \cite[Lemma 6.2]{FI}, we see that $\{u_{n_{k}}\}$ converges to $u$ in $W^{1,p}_{0}(\Omega)$. \end{proof} \begin{proof}[Proof of theorem \ref{theorem2}] Using the Mountain-pas theorem with the Cerami condition, by Lemmas \ref{lemma2}, \ref{lemma1} and \ref{lemmcc1}, we obtain a non-trivial weak solution for the problem \eqref{eqP}. \end{proof} \begin{example} \label{ex12b}\rm Let $N=5$, $p=3$, $r = 4$, $\alpha >0$, $\Omega =\{x\in \mathbb{R}^{5}: \|x\| < 1\}$, \begin{gather*} \omega_{0}(x)=|x|^{-1/30}\cos(16|x|) \quad \forall x\in\Omega,\\ \omega_1(x)=(\frac{1}{2}-\|x\|^{2})^{2}(1-\|x\|^{2})^{-7/6}\quad \forall x\in\Omega,\\ \varphi_{0}(t) = \begin{cases} |t|^{r-2}t(1-|t|)& \text{if }|t| \le 1, \\ 0 &\text{if }|t| \in\mathbb{R}\setminus [-1,1], \end{cases} \\ \varphi_1(t)= |t|^{p-2}t\varphi_1 (t)\log(1+|t|)\quad h\forall t\in\mathbb{R},\\ f(x,t)=\omega_{0}(x)^{r-1}\varphi_{0}(t)+\omega_1(x)^{r-1}\varphi_1(t)\quad \forall (x,t)\in \Omega\times \mathbb{R}. \end{gather*} Let $\omega =|\omega_{0}| +\omega_1$, $C=1$, $d(x)=|x|^{-\frac{1}{30}}$, $d_1(x)= -d(x)$ and $d_{2}(x)=|x|^{-\frac{1}{30}}$ for every $x$ in $\Omega$. We see that $d \in L^{1}(\Omega)$, $d_1 \in L^{\frac{N}{p}}(\Omega)$ and $d_{2} \in L^{1}(\Omega)$. By Examples \ref{ex10} and \ref{ex11}, $\omega$ is in $\mathcal{K}_{p,r}$. Thus $f$ satisfies conditions (A1)--(A5). Since $\lim_{|x|\to 0}\omega_{0}(x) =\infty$ and $\lim_{|x|\to \frac{1}{2}}\omega_1(x) =0$, the convergence in (A4) and (A5) are not uniform on $\Omega$. We have $ \frac{f(x,t)}{|t|^{p-2}t}=\omega_1(x)(|t|-1)\log(1+|t|)$ for every $t\in[-2,2]\setminus [-1,1]$ and $ \frac{f(x,t)}{|t|^{p-2}t}=\omega_1(x)\log(1+|t|)$ for every $t\in\mathbb{R}\setminus [-2,2]$. Thus $f$ satisfies (A6). Therefore we can apply Theorem \ref{theorem2} to $f$ with $C=1$ respectively. Since $\omega^{r -1}(x)\ge(1-\|x\|^{2})^{-\frac{21}{16}}$ for every $x$ in $\Omega$, $\omega^{r -1}$ is not integrable on $\Omega$. 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