\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2014 (2014), No. 170, pp. 1--13.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2014 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2014/170\hfil Nonlinear elliptic problem of $2$-$q$-Laplacian type] {Nonlinear elliptic problem of $2$-$q$-Laplacian type with asymmetric nonlinearities} \author[D. Yang, C. Bai \hfil EJDE-2014/170\hfilneg] {Dandan Yang, Chuanzhi Bai} % in alphabetical order \address{Dandan Yang \newline School of Mathematical Science, Huaiyin Normal University, Huaian, Jiangsu 223300, China} \email{ydd423@sohu.com} \address{Chuanzhi Bai \newline School of Mathematical Science, Huaiyin Normal University, Huaian, Jiangsu 223300, China} \email{czbai@hytc.edu.cn} \thanks{Submitted July 9, 2014. Published August 11, 2014.} \subjclass[2000]{35J60, 35B38} \keywords{Quasilinear elliptic equations with $q$-Laplacian; critical exponent; \hfill\break\indent asymmetric nonlinearity; weak solution} \begin{abstract} In this article, we study the nonlinear elliptic problem of $2$-$q$-Laplacian type \begin{gather*} - \Delta u - \mu \Delta_q u = - \lambda |u|^{r-2} u + a u + b (u^+)^{\theta-1} \quad\text{in } \Omega, \\ u = 0 \quad\text{on } \partial \Omega, \end{gather*} where $\Omega \subset \mathbb{R}^N $ is a bounded domain. For $a$ is between two eigenvalues, we show the existence of three nontrivial solutions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{corollary}[theorem]{Corollary} \allowdisplaybreaks \section{Introduction} In this article, we are interested in finding the multiple nontrivial weak solutions to the nonlinear elliptic problem of $2$-$q$-Laplacian type, \begin{equation} \begin{gathered} - \Delta u - \mu \Delta_q u = - \lambda |u|^{r-2} u + a u + b (u^+)^{\theta-1} \quad \text{in } \Omega, \\ u = 0 \quad \text{on } \partial \Omega, \end{gathered} \label{e1.1} \end{equation} where $\Omega \subset \mathbb{R}^N $ is a bounded domain with samooth boundary $\partial \Omega$, $\lambda, \mu > 0$ are two parameters, $N > 2$, $1 < \min \{q, r\} \le \max \{q, r\} <2 < \theta \le2^* = \frac{2 N}{N - 2}$, $a \in \mathbb{R}$, $b > 0$, and $u^+ = \max \{u, 0\}$. $\Delta_q u = \operatorname{div}(|\nabla u|^{q-2} \nabla u)$ is the $q$-Laplacian of $u$. Paiva and Presoto \cite{p1} studied the semilinear elliptic problem with asymmetric nonlinearities, \begin{equation} \begin{gathered} - \Delta u = - \lambda |u|^{q-2} u + a u + b (u^+)^{p-1} \quad \text{in } \Omega, \\ u = 0 \quad\text{on } \partial \Omega. \end{gathered} \label{e1.2} \end{equation} Where $N \ge 3$, $1 < q < 2 < p \le 2^*$, $a \in \mathbb{R}$, $b > 0$ and $\lambda$ is a positive parameter. Problem \eqref{e1.2} is also closely related to the class of superlinear Ambrosetti-Prodi problems \cite{f1}, \begin{equation} - \Delta u = a u + (u^+)^p + f(x) \quad \text{in } \Omega. \label{e1.3} \end{equation} Further results for problem \eqref{e1.3} can be found in \cite{c1,c2,m2,r1} and references cited therein. Marano and Papageorgiou \cite{m1} obtained the existence of three solutions of the ($p,q$)-Laplacian problem \begin{equation} \begin{gathered} - \Delta_p u - \mu \Delta_q u = f(x, u) \quad \text{in } \Omega, \\ u = 0 \quad\text{on } \partial \Omega, \end{gathered}\label{e1.4} \end{equation} by using variational methods and truncation arguments. Nonlinear elliptic problems involving the $p$-$q$-Laplacian operator is an active are of research; see \cite{h1,l1,p2,s1,y1,y2} and the references therein. Motivated by the above works, we shall extend the results of problem \eqref{e1.2} to problem \eqref{e1.1}. By using variational methods, we obtain three solutions to \eqref{e1.1}. We say that $g$ is asymmetric when $g$ satisfies the Ambrosetti-Prodi type condition \begin{align*} g_- := \lim_{t \to - \infty} \frac{g(t)}{t} < \lambda_k < g_+ := \lim_{t \to + \infty} \frac{g(t)}{t}. \end{align*} Since problem \eqref{e1.1} involves $- \Delta$ and $- \Delta_q$, the arguments will be more complicated, and more analysis and estimates are needed. The eigenvalue problem of the Laplacian, in $\Omega \subset \mathbb{R}^N$, has the form \begin{equation} - \Delta u = \lambda u \quad \text{in } \ H_0^1(\Omega).\label{e1.5} \end{equation} By the Ljusternik-Schnirelman principle it is well known that there exists a nondecreasing sequence of nonnegative eigenvalues $0 < \lambda_1 < \lambda_2 \le \dots \le \lambda_j \le \dots $ and a correspondent eigenfunctions $\varphi_j$. Also, the first eigenvalue $\lambda_1$ is simple and the eigenfunctions associated with $\lambda_1$ do not change sign. Now we are ready to state our main results. \begin{theorem}\label{thm1.1} Let $N \ge 3$, $1 < \min \{q,r\} \le \max \{q, r\} < 2 < \theta < 2^*$ and $\lambda_k < a <\lambda_{k+1}$. Then, for $\lambda > 0$ and $\mu > 0$ small enough, problem \eqref{e1.1} has at least three nontrivial solutions. \end{theorem} \begin{theorem}\label{thm1.2} Let $N \ge 4$, $1 < \min \{q,r\} \le \max \{q, r\} < 2 < \theta = 2^*$ and $\lambda_k < a < \lambda_{k+1}$. Then, for $\lambda > 0$ and $\mu > 0$ small enough, problem \eqref{e1.1} has at least three nontrivial solutions. \end{theorem} This article is organized as follows. In Section 2, we show some geometric conditions to establish the Mountain-Pass levels and give a technical lemma which is crucial in the proof of our main results. In Section 3, we establish the existence of three nontrivial solutions for the nonlinear elliptic problem \eqref{e1.1}. \section{Preliminaries} In this article, $\|\cdot\|_p$ and $|\cdot|_p$ denote the norms on $W_0^{1, p}(\Omega)$ and $L^p(\Omega)$, respectively; \[ \|u\|_p = \Big(\int_{\Omega} |\nabla u|^p dx\Big)^{1/p}, \quad |u|_p = \Big(\int_{\Omega} |u|^p dx\Big)^{1/p}. \] For convenience, we substitute $\|\cdot\|$ for $\|\cdot\|_2$. The best Sobolev constant $S$ of the embedding $H_0^1(\Omega) \hookrightarrow L^{2^*}(\Omega)$ is denoted by $$ S = \inf _{u \in H_0^1(\Omega) \setminus \{0\}} \frac{\|u\|^2}{|u|_{2^*}^2}. $$ It is known that $S$ is independent of $\Omega$ and is never achieved except when $\Omega = \mathbb{R}^N$ (see \cite{t1}). Consider the energy functional $I_{\lambda, \mu}$ defined on $H_0^1(\Omega)$ given by \begin{equation} I_{\lambda, \mu}(u) = \frac{1}{2} \|u\|^2 + \frac{\mu}{q} \|u\|_q^q + \frac{\lambda}{r} \int_{\Omega} |u|^r dx - \frac{a}{2} \int_{\Omega} |u|^2 dx - \frac{b}{\theta} \int_{\Omega} (u^+)^{\theta} dx.\label{e2.1} \end{equation} It is easy to know that $I_{\lambda, \mu}$ is of class $\mathcal{C}^2$ and there exists a one to one correspondence between the weak solutions of \eqref{e1.1} and the critical points of $I_{\lambda, \mu}$ on $H_0^1(\Omega)$. By a weak solution of \eqref{e1.1} we mean that $u \in H_0^1(\Omega)$ satisfying \begin{align*} \langle I_{\lambda, \mu}'(u), v\rangle &= \int_{\Omega} [\nabla u \nabla v + \mu |\nabla u|^{q-2} \nabla u \nabla v] dx + \lambda \int_{\Omega} |u|^{r-2} u v dx \\ &\quad - a \int_{\Omega} u v dx - b \int_{\Omega} (u^+)^{\theta-1} v dx = 0 \end{align*} for all $v \in H_0^1(\Omega)$. Denote by $\varphi_i$ a normalized eigenvector relative to eigenvalue $\lambda_i$ of \eqref{e1.5}. Let $V_k = \langle \varphi_1, \dots , \varphi_k \rangle$ and $W_k = V_k^{\bot}$. Without loss of generality, we suppose $0 \in \Omega$, and $m \in \mathbb{N}$ large enough so that $B_{2/m} \subset \Omega$, where $B_{2/m}$ denotes the ball of radius $2/m$ with center in 0. Consider the functions introduced in \cite{g1}, \[ \zeta_m(x) = \begin{cases} 0 & \text{if } x \in B_{1/m}, \\ m|x| - 1 & \text{if } x \in A_m = B_{2/m} \setminus B_{1/m}, \\ 1 & \text{if } x \in \Omega \setminus B_{2/m}. \end{cases} \] Set $\varphi_i^m = \zeta_m \varphi_i$, $$ V_k^m = \langle \varphi_1^m, \varphi_2^m, \dots , \varphi_k^m \rangle $$ and $W_k^m = (V_k^m)^{\bot}$. For each $m \in \mathbb{N}$, define a positive cut-off function $\eta \in \mathcal{C}_c^{\infty} (B_{1/m})$ such that $\eta \equiv 1$ in $B_{1/2m}$, $\eta \le 1$ in $B_{1/m}$ and $\|\nabla \eta\|_{\infty} \le 4 m$; take $\varphi_{k+1}^m = \eta \varphi_{k+1}$. Then \begin{equation} \operatorname{supp} u \cap \operatorname{supp} \varphi_{k+1}^m = \emptyset\label{e2.2} \end{equation} whenever $u \in V_k^m$. By \cite{g1}, it is easy to check the following Lemma. \begin{lemma} \label{lem2.1} As $m \to \infty$ we have $$ \varphi_i^m \to \varphi_i \quad \text{in } H_0^1(\Omega) \quad \text{and} \quad \max _{u \in V_k^m :\int_{\Omega} |u|^2 = 1} \|u\|^2 \le \lambda_k + c_k m^{2-N}. $$ \end{lemma} \begin{corollary}\label{cor2.2} For $m$ large enough \begin{equation} V_k^m \oplus W_k = H_0^1.\label{e2.3} \end{equation} \end{corollary} As an easy consequence of Lemma 2.1 we have the following decomposition of $H_0^1$. \begin{lemma}\label{lem2.3} Assume $\lambda_1 < a$, $1 < \min \{q, r\} \le \max \{q, r\} < 2 < \theta \le 2^*$ and $\lambda, \mu > 0$. Then every {\rm (PS)} sequence of $I_{\lambda, \mu}$ is bounded. \end{lemma} \begin{proof} Suppose $\{u_n\} \subset H_0^1(\Omega) $ is a (PS) sequence of $I_{\lambda, \mu}$; i.e., it satisfies \begin{gather} \Big|\frac{1}{2} \|u_n\|^2 + \frac{\mu}{q} \|u_n\|_q^q + \frac{\lambda}{r} \int_{\Omega} |u_n|^r dx - \frac{a}{2} \int_{\Omega} |u_n|^2 dx - \frac{b}{\theta} \int_{\Omega} (u_n^+)^{\theta} dx\Big| \le C,\label{e2.4} \\ \begin{aligned} &\Big|\int_{\Omega} [\nabla u_n \nabla v + \mu |\nabla u_n|^{q-2} \nabla u_n \nabla v] dx + \lambda \int_{\Omega} |u_n|^{r-2} u_n v dx \\ &- a \int_{\Omega} u_n v dx - b \int_{\Omega} (u_n^+)^{\theta-1} v dx\Big| \le \epsilon_n \|v\|, \quad \forall v \in H_0^1(\Omega), \end{aligned}\label{e2.5} \end{gather} where $\epsilon_n \to 0$ as $n \to \infty$. By \eqref{e2.4} and \eqref{e2.5}, we obtain \begin{equation} \begin{aligned} &C + \epsilon_n \|u_n\|\\ & \ge \big|I_{\lambda}(u_n) - \frac{1}{2} \langle I_{\lambda}'(u_n), u_n\rangle \big| \\ & = \Big|\big(\frac{\mu}{q} - \frac{\mu}{2}\big) \|u_n\|_q^q + \big(\frac{\lambda}{r} - \frac{\lambda}{2}\big) \int_{\Omega} |u_n|^r dx + \big(\frac{b}{2} - \frac{b}{\theta}\big) \int_{\Omega} (u_n^+)^{\theta} dx\Big| \\ & \ge \big(\frac{b}{2} - \frac{b}{\theta}\big) \int_{\Omega} (u_n^+)^{\theta} dx. \end{aligned}\label{e2.6} \end{equation} Thus, we have \begin{equation} \int_{\Omega} (u_n^+)^{\theta} dx \le C + \epsilon_n \|u_n\|. \label{e2.7} \end{equation} Moreover, by H\"older inequality, we have \begin{equation} \int_{\Omega} (u_n^+)^2 dx \le |\Omega|^{\frac{\theta-2}{\theta}} \Big(\int_{\Omega}(u_n^+)^{\theta} dx\Big)^{2/\theta}. \label{e2.8} \end{equation} On the other hand, by \eqref{e2.5} we have \begin{equation} |\langle I_{\lambda, \mu}'(u_n), u_n^- \rangle| = \big|\|u_n^-\|^2 + \mu \|u_n^-\|_q^q + \lambda |u_n^-|_r^r - a |u_n^-|_2^2\big| \le \epsilon_n \|u_n^-\|, \label{e2.9} \end{equation} with $u^- = \max \{-u, 0\}$. It follows from \eqref{e2.4}, \eqref{e2.7}, \eqref{e2.8} and \eqref{e2.9} that \begin{equation} \begin{aligned} \frac{1}{2} \|u_n^+\|^2 &\le \big(\frac{\mu}{2} -\frac{\mu}{q}\big) \|u_n^-\|_q^q + \big(\frac{\lambda}{2} -\frac{\lambda}{r}\big)\int_{\Omega} |u_n|^r\\ &\quad + \frac{a}{2}\int_{\Omega} (u_n^+)^2 dx + \frac{b}{\theta} \int_{\Omega}(u_n^+)^{\theta} dx + \frac{1}{2} |\langle I_{\lambda, \mu}'(u_n), u_n^- \rangle| + C \\ &\le \frac{a}{2} \int_{\Omega} (u_n^+)^2 dx + \frac{b}{\theta} \int_{\Omega} (u_n^+)^{\theta} dx + \epsilon_n \|u_n^-\| + C \\&\le \epsilon_n \|u_n\| + \epsilon_n \|u_n^-\| + C. \end{aligned} \label{e2.10} \end{equation} Firstly, we show that $(u_n^+)$ is bounded in $H_0^1(\Omega)$. Suppose by contradiction that $\|u_n^+\| \to \infty$, by \eqref{e2.10}, we know that $(u_n^-)$ is also unbounded. Let $w_n = u_n/\|u_n\|$. Since $\{w_n\}$ is bounded in $H_0^1(\Omega)$, there exists $w \in H_0^1(\Omega)$ such that \begin{gather*} w_n \rightharpoonup w \quad \text{in } H_0^1(\Omega), \\ w_n \to w \quad \text{in } L^s, \; \forall 1 \le s < 2^* , \\ w_n \to w \quad\text{a.e. in } \Omega. \end{gather*} From \eqref{e2.10}, there exists $\sigma > 0$ satisfying \begin{equation} \|u_n^-\| \ge \sigma \|u_n^+\|^2\label{e2.11} \end{equation} whenever $n$ is large. Notice that $$ w_n^+ = \frac{u_n^+}{\|u_n\|} = \frac{u_n^+}{(\|u_n^+\|^2 + \|u_n^-\|^2)^{1/2}} \le \frac{u_n^+}{(\|u_n^+\|^2 + \sigma^2 \|u_n^+\|^4)^{1/2}}, $$ which implies that $w \le 0$. Furthermore, by $$ w_n^- = \frac{u_n^-}{\|u_n\|} = \frac{u_n^-}{(\|u_n^+\|^2 + \|u_n^-\|^2)^{1/2}} = \frac{u_n^-}{\|u_n^-\|} \cdot \frac{\|u_n^-\|}{(\|u_n^+\|^2 + \|u_n^-\|^2)^{1/2}} $$ and \eqref{e2.11}, we obtain $\|w_n^-\| \to 1$. Hence, by \eqref{e2.9}, \begin{equation} - \lambda \frac{|u_n^-|_r^r}{\|u_n^-\|^2} + \mu \frac{\|u_n^-\|_q^q}{\|u_n^-\|^2} + a \frac{|u_n^-|^2}{\|u_n^-\|^2} \to 1.\label{e2.12} \end{equation} Recalling that $q, r < 2$, we obtain \begin{gather} \frac{\|u_n^-\|_q^q}{\|u_n^-\|^2} \le |\Omega|^{\frac{2-q}{2}} \|u_n^-\|^{q-2} \to 0,\label{e2.13} \\ \frac{|u_n^-|_r^r}{\|u_n^-\|^2} \le |\Omega|^{\frac{2^* - r}{2^*}} S^{- \frac{r}{2 2^*}} \|u_n^-\|^{\frac{r}{2^*}-2} \to 0.\label{e2.14} \end{gather} Moreover, by \eqref{e2.11} and $\|w_n^-\| \to 1$, we have $$ \frac{u_n^-}{\|u_n^-\|} - \frac{u_n^-}{\|u_n\|} = \frac{u_n^-}{\|u_n\|} \big(\frac{\|u_n\|}{\|u_n^-\|} - 1\big) \to 0 \quad \text{in } H_0^1(\Omega). $$ Thus we may exchange $\|u_n^-\|$ for $\|u_n\|$ in \eqref{e2.12}, and substituting \eqref{e2.13} and \eqref{e2.14} into it, we obtain $|w_n^-| \to 1/\sqrt{a}$, then $w\neq 0$. Taking $v = \varphi_1$ in \eqref{e2.5}, one has \begin{align*} &\int_{\Omega} [\nabla w_n \nabla \varphi_1 dx + \mu \frac{\|u_n\|_q^{q-1}}{\|u_n\|} \int_{\Omega} |\nabla w_n|^{q-2} \nabla w_n \nabla \varphi_1 dx \\ &+ \frac{\lambda}{\|u_n\|} \int_{\Omega} |u_n|^{r-2} u_n \varphi_1 dx - a \int_{\Omega} w_n \varphi_1 dx - \frac{b}{\|u_n\|} \int_{\Omega} (u_n^+)^{\theta-1} \varphi_1 dx \to 0; \end{align*} that is, \begin{equation} \begin{aligned} &(\lambda_1 - a) \int_{\Omega} w_n \varphi_1 dx + \mu \frac{\|u_n\|_q^{q-1}}{\|u_n\|} \int_{\Omega} |\nabla w_n|^{q-2} \nabla w_n \nabla \varphi_1 dx \\ & + \frac{\lambda}{\|u_n\|} \int_{\Omega} |u_n|^{r-2} u_n \varphi_1 dx - \frac{b}{\|u_n\|} \int_{\Omega} (u_n^+)^{\theta-1} \varphi_1 dx \to 0.\end{aligned} \label{e2.15} \end{equation} Since the second, the third and the fourth term above approach zero, it follows that $$ (\lambda_1 - a) \int_{\Omega} w \varphi_1 dx = 0, $$ which is a contradiction, as $w \le 0$, $w \neq 0$ and $\lambda_1 < a$, so that $(u_n^+)$ is bounded. Finally, assume that $\|u_n\| \to \infty$ and $\|u_n^+\| \le C$ for all $n \in \mathbb{N}$. Taking $v = w_n$ in \eqref{e2.5}, by \eqref{e2.13} and $$ \frac{1}{\|u_n\|} \int_{\Omega} (u_n^+)^{\theta} dx \to 0, $$ for $\theta \le 2^*$, we obtain $a |w_n|_2^2 \to 1$, so that $w_n \to w$ in $L^2(\Omega)$ with $w \neq 0$. Then by \eqref{e2.5} we obtain $$ \int_{\Omega} \nabla w \nabla v dx - a \int_{\Omega} w v dx = 0 \quad {\rm for \ all} \ v \in H_0^1(\Omega), $$ with $w \neq 0$ and $w \le 0$, which is a contradiction, as $a$ is not the first eigenvalue. Hence, we conclude that $\{u_n\}$ must be bounded in $ H_0^1(\Omega)$. \end{proof} In the subcritical case, $1 \le \theta < 2^*$, we can easily know according to the lemma above, $I_{\lambda, \mu}$ satisfies the (PS) condition at every level. \begin{lemma} \label{lem2.4} Let $\lambda_1 < a$ and $\theta =2^*$. For each $\lambda, \mu > 0$, $I_{\lambda, \mu}$ satisfies the {\rm (PS)} condition at level $c$ with $c < \frac{1}{N} b^{\frac{2-N}{2}} S^{N/2}$. \end{lemma} \begin{proof} Let $\{u_n\} \subset H_0^1(\Omega)$ be a sequence satisfying \begin{equation} I_{\lambda, \mu}(u_n) \to c \quad \text{and} \quad |\langle I_{\lambda, \mu}'(u_n), v \rangle| \le \epsilon_n \|v\|_p, \quad \forall v \in H_0^1(\Omega), \label{e2.16} \end{equation} with $\epsilon_n \to 0$ as $n \to \infty$. By Lemma \ref{lem2.3} we obtain that $\{u_n\}$ is bounded. Thus, by passing to a subsequence, we have \begin{equation} \begin{gathered} u_n \rightharpoonup u \quad \text{in } H_0^1(\Omega), \\ u_n \to u \quad \text{in } L^s, \; \forall 1 \le s < 2^*,\\ u_n \to u \quad\text{a.e. in } \Omega. \end{gathered}\label{e2.17} \end{equation} Since $\{u_n^+\}$ is bounded in $H_0^1(\Omega)$, from the Gagliardo-Nirenberg inequality it follows that $\{u_n^+\}$ is also bounded in $L^{2^*}$. By passing to a subsequence again, we have $u_n^+ \rightharpoonup u^+$ in $L^{2^*}$. Hence, we obtain by \cite[Lemma 2.3]{m2} that \begin{equation} \begin{gathered} - \Delta u - \mu \Delta_q u = - \lambda |u|^{r-2} u + a u + b (u^+)^{2^*-1}, \quad \text{in } \Omega \\ u = 0 \quad \text{on } \partial \Omega, \end{gathered} \label{e2.18} \end{equation} Thus, by \eqref{e2.18} we have \begin{equation} I_{\lambda, \mu}(u) = \big(\frac{\mu}{q} - \frac{\mu}{2}\big) \|u\|_q^q + \big(\frac{\lambda}{r} - \frac{\lambda}{2}\big) \int_{\Omega} |u|^r dx + \big(\frac{b}{2} - \frac{b}{2^*}\big) \int_{\Omega} (u^+)^{2^*} dx \ge 0. \label{e2.19} \end{equation} Set $w_n = u_n - u$. It is easy to check that \begin{equation} |u_n^+ - u^+|_s^s \le |(u_n - u)^+|_s^s = |w_n^+|_s^s, \quad 1 \le s \le 2^*. \label{e2.20} \end{equation} By \eqref{e2.16} and the Brezis-Lieb Lemma, we have \begin{align*} &\|w_n\|^2 + \mu \|w_n\|_q^q + \lambda |w_n|_r^r - a |w_n|_2^2 - b |u_n^+ - u^+|_{2^*}^{2^*}\\ & = \|u_n\|^2 - \|u\|^2 + \mu (\|u_n\|_q^q - \|u\|_q^q) + \lambda (|u_n|_r^r - |u|_r^r)\\ &\quad - a (|u_n|_2^2 - |u|_2^2) - b \big(|u_n^+|_{2^*}^{2^*} - |u^+|_{2^*}^{2^*}\big) + o_n(1)\\ &= \langle I_{\lambda, \mu}'(u_n), u_n \rangle - \langle I_{\lambda, \mu}'(u), u \rangle + o(1), \end{align*} which implies that \begin{equation} \lim_{n \to \infty} \big[\|w_n\|^2 + \mu \|w_n\|_q^q + \lambda |w_n|_r^r - a |w_n|_2^2 - b |u_n^+ - u^+|_{2^*}^{2^*}\big] = 0.\label{e2.21} \end{equation} Moreover, by \eqref{e2.17} we have $w_n \to 0$ in $L^r$ and $L^2$. Thus, we have from \eqref{e2.20} and \eqref{e2.21} that \begin{equation} \|w_n\|^2 + \mu \|w_n\|_q^q = b |u_n^+ - u^+|_{2^*}^{2^*} + o(1) \le b |w_n^+|_{2^*}^{2^*} + o(1).\label{e2.22} \end{equation} Without loss of generality, we assume that \begin{equation} \|w_n\|^2 = d + o(1), \quad \|w_n\|_q^q = h + o(1). \label{e2.23} \end{equation} By \eqref{e2.22}, \eqref{e2.23} and Sobolev inequality, we obtain \begin{equation} d \ge S \Big(\frac{d+\mu h}{b}\Big)^{2/2^*} \ge S b^{-2/2^*} d^{2/2^*} .\label{e2.24} \end{equation} If $d=0$, then we complete the proof. Otherwise, \eqref{e2.24} implies that \begin{equation} d \ge S^{N/2} b^{\frac{2-N}{2}}. \label{e2.25} \end{equation} Then by \eqref{e2.16}, \eqref{e2.19} and the Brezis-Lieb Lemma, we conclude \begin{equation} \begin{aligned} c &\ge c - I_{\lambda, \mu}(u) = I_{\lambda, \mu}(u_n) - I_{\lambda, \mu}(u) + o(1)\\ & = \frac{1}{2} \left(\|u_n\|^2 - \|u\|^2\right) + \frac{\mu}{q} \left(\|u_n\|_q^q - \|u\|_q^q\right)\\ & \quad + \frac{\lambda}{r} \left(|u_n|_r^r - |u|_r^r\right) - \frac{a}{2} \left(|u_n|_2^2 - |u|_2^2\right) - \frac{b}{2^*} \big(|u_n^+|_{2^*}^{2^*} - |u^+|_{2^*}^{2^*} \big) + o(1)\\ &= \frac{1}{2} \|w_n\|^2 + \frac{\mu}{q} \|w_n\|_q^q + \frac{\lambda}{r} |w_n|_r^r - \frac{a}{2} |w_n|_2^2 - \frac{b}{2^*} |u_n^+ - u^+|_{2^*}^{2^*} + o(1). \end{aligned} \label{e2.26} \end{equation} Let $n \to \infty$ in \eqref{e2.26}, we obtain by \eqref{e2.22}, \eqref{e2.23}, \eqref{e2.25} and $w_n \to 0$ in $L^r$ and $L^2$ that \begin{align*} c & \geq \frac{d}{2} + \frac{\mu h}{q} - \frac{d + \mu h}{2^*}\\ &= \big(\frac{1}{2} - \frac{1}{2^*}\big) d + \big(\frac{\mu}{q} - \frac{\mu}{2^*}\big) h\\ &\ge \big(\frac{1}{2} - \frac{1}{2^*}\big) d \\ &\ge \frac{1}{N} S^{N/2} b^{\frac{2-N}{2}}, \end{align*} which is a contradiction. \end{proof} \section{Main result} Firstly, we consider the existence of the nonnegative solution of \eqref{e1.1} . Define the functional $I_{\lambda, \mu}^+ : H_0^1(\Omega) \to \mathbb{R}$ as follows \begin{equation} I_{\lambda, \mu}^+(u) = \frac{1}{2} \|u\|^2 + \frac{\mu}{q} \|u\|_q^q + \frac{\lambda}{r} \int_{\Omega} (u^+)^r dx - \frac{a}{2} \int_{\Omega} (u^+)^2 dx - \frac{b}{\theta} \int_{\Omega} (u^+)^{\theta} dx. \label{e3.1} \end{equation} It follows that $I_{\lambda, \mu}^+ \in C^1$ and the critical points $u_+$ of $I_{\lambda, \mu}^+$ satisfy $u_+ \ge 0$ and so are critical points of $I_{\lambda, \mu}$ as well, actually, $(I_{\lambda, \mu}^+)'(u_+)[(u_+)^-] = - \|(u_+)^-\|^2 - \mu \|(u_+)^-\|_q^q = 0$. Similar to the proofs of Lemma \ref{lem2.3} and Lemma \ref{lem2.4}, we can show that $I_{\lambda, \mu}^+$ satisfies the (PS) condition. \begin{lemma}\label{lem3.1} Let $2 < \theta \le 2^*$. If $\lambda, \mu > 0$, then $I_{\lambda, \mu}^+$ satisfies the {\rm (PS)} condition at level $c$ with $c < \frac{1}{N} S^{N/2} b^{\frac{2-N}{2}}$. \end{lemma} \begin{lemma}\label{lem3.2} The trivial solution $u \equiv 0$ is a local minimizer for $I_{\lambda, \mu}^+$, for all $\lambda, \mu > 0$. \end{lemma} \begin{proof} It suffices to show that $0$ is a local minimizer of $I_{\lambda, \mu}^+$ in the topology (see \cite{b1}). For $u \in C_0^1(\overline{\Omega})$, we have \begin{align*} I_{\lambda, \mu}^+(u) &= \frac{1}{2} \|u\|^2 + \frac{\mu}{q} \|u\|_q^q + \frac{\lambda}{r} \int_{\Omega} (u^+)^r dx - \frac{a}{2} \int_{\Omega} (u^+)^2 dx - \frac{b}{\theta} \int_{\Omega} (u^+)^{\theta} dx \\ &\ge \frac{\lambda}{r} \int_{\Omega} (u^+)^r dx - \frac{a}{2} \int_{\Omega} (u^+)^2 dx - \frac{b}{\theta} \int_{\Omega} (u^+)^{\theta} dx \\ &\ge \Big(\frac{\lambda}{r} - \frac{a}{2} |u|_{C^0}^{2-r} - \frac{b}{\theta} |u|_{C^0}^{\theta-r}\Big) \int_{\Omega} (u^+)^r dx \ge 0 \end{align*} whenever \[ \frac{a}{2} |u|_{C^0}^{2-r} + \frac{b}{\theta} |u|_{C^0}^{\theta-r} \le \frac{\lambda}{r}. \] \end{proof} \begin{lemma}\label{lem3.3} There exists $t_0 > 0$ such that $I_{\lambda, \mu}^+(t_0 \varphi_1) \le 0$, for all $\lambda, \mu$ in a bounded set. \end{lemma} \begin{proof} Let $\varphi_1$ be the positive eigenfunction associated to $\lambda_1$, for $t > 0$, we have \begin{align*} I_{\lambda, \mu}^+(t \varphi_1) & = \frac{t^2}{2} \|\varphi_1\|^2 + \frac{t^q \mu}{q} \|\varphi_1\|_q^q + \frac{t^r \lambda}{r} \int_{\Omega} \varphi_1^r dx - \frac{t^2 a}{2} \int_{\Omega} \varphi_1^2 dx - \frac{t^{\theta} b}{\theta} \int_{\Omega} \varphi_1^{\theta} dx\\ &= \frac{t^2}{2}(\lambda_1 - a) \int_{\Omega} \varphi_1^2 dx + \frac{t^q \mu}{q} \|\varphi_1\|_q^q + \frac{t^r \lambda}{r} \int_{\Omega} \varphi_1^r dx - \frac{t^{\theta} b}{\theta} \int_{\Omega} \varphi_1^{\theta} dx \end{align*} Since $\lambda_1 < a$ and $q, r < 2 < \theta$, there exists a choice of $t_0 > 0$ such that $I_{\lambda, \mu}^+(t_0 \varphi_1) \le 0$ for $\lambda, \mu$ in a bounded set. \end{proof} Define $$ c_{\lambda, \mu}^+ = \inf_{\gamma \in \Gamma^+} \sup _{t \in [0, 1]} I_{\lambda, \mu}^+(\gamma(t)), $$ where $$ \Gamma^+ = \{\gamma \in \mathcal{C}([0, 1], \gamma(0) = 0, \gamma(1) = t_0 \varphi_1\}. $$ On the other hand, by the proof of Lemma \ref{lem3.3}, we obtain $$ I_{\lambda, \mu}^+(t \varphi_1) \le \frac{t^q \mu}{q} \|\varphi_1\|_q^q + \frac{t^r \lambda}{r} \int_{\Omega} \varphi_1^r dx. $$ Then, if $\lambda$ and $\mu$ are small enough, $c_{\lambda, \mu}^+ < \frac{1}{N} S^{N/2} b^{\frac{2-N}{2}}$, consequently, by means of the Mountain Pass Theorem, $c_{\lambda, \mu}^+$ is a critical value of $I_{\lambda, \mu}^+$. Thus, we have the following result. \begin{lemma}\label{lem3.4} Let $N > 2$, $1 < \min \{q, r\} \le \max \{q, r\} < 2 < \theta \le 2^*$ and $\lambda_1 < a$. If $\lambda, \mu$ are small enough, then \eqref{e1.1} has at least a nontrivial positive solution. \end{lemma} To obtain the negative solution, consider the functional $I_{\lambda, \mu}^- : H_0^1(\Omega) \to \mathbb{R}$ given by \begin{equation} I_{\lambda, \mu}^-(u) = \frac{1}{2} \|u\|^2 + \frac{\mu}{q} \|u\|_q^q + \frac{\lambda}{r} \int_{\Omega} (u^-)^r dx - \frac{a}{2} \int_{\Omega} (u^-)^2 dx. \label{e3.2} \end{equation} Again, $I_{\lambda, \mu}^- \in C^1$ and the critical points $u_-$ of $I_{\lambda, \mu}^-$ satisfy $u_- \le 0$ and so are critical points of $I_{\lambda, \mu}$ as well. We will apply once again the mountain pass theorem to obtain a critical point of $I_{\lambda, \mu}^-$. \begin{lemma}\label{lem3.5} The trivial solution $u \equiv 0$ is a local minimizer for $I_{\lambda, \mu}^-$, for all $\lambda, \mu > 0$. \end{lemma} \begin{proof} It suffices to show that $0$ is a local minimizer of $I_{\lambda, \mu}^-$ in the topology. For $u \in C_0^1(\overline{\Omega})$, we have \begin{align*} I_{\lambda, \mu}^-(u) &= \frac{1}{2} \|u\|^2 + \frac{\mu}{q} \|u\|_q^q + \frac{\lambda}{r} \int_{\Omega} (u^-)^r dx - \frac{a}{2} \int_{\Omega} (u^-)^2 dx \\ &\ge \frac{\lambda}{r} \int_{\Omega} (u^-)^r dx - \frac{a}{2} \int_{\Omega} (u^-)^2 dx \\ &\ge \big(\frac{\lambda}{r} - \frac{a}{2} |u|_{C^0}^{2-r} \big) \int_{\Omega} (u^-)^r dx \ge 0 \end{align*} whenever $\frac{a}{2} |u|_{C^0}^{2-r} \le \lambda/r$. \end{proof} \begin{lemma}\label{lem3.6} There exists $t_0 > 0$ such that $I_{\lambda, \mu}^-(- t_0 \varphi_1) \le 0$, for all $\lambda, \mu$ in a bounded set. \end{lemma} \begin{proof} For $t > 0$, we have \begin{align*} I_{\lambda, \mu}^-(- t \varphi_1) &= \frac{t^2}{2} \|\varphi_1\|^2 + \frac{t^q \mu}{q} \|\varphi_1\|_q^q + \frac{t^r \lambda}{r} \int_{\Omega} \varphi_1^r dx - \frac{t^2 a}{2} \int_{\Omega} \varphi_1^2 dx \\ & = \frac{t^2}{2}(\lambda_1 - a) \int_{\Omega} \varphi_1^2 dx + \frac{t^q \mu}{q} \|\varphi_1\|_q^q + \frac{t^r \lambda}{r} \int_{\Omega} \varphi_1^r dx. \end{align*} Since $\lambda_1 < a$ and $r, q < 2$, there exists a choice of $t_0> 0$ which proves the lemma. \end{proof} As in the nonnegative solution case, we obtain a critical value $$ c_{\lambda, \mu}^- = \inf_{\gamma \in \Gamma^-} \sup _{t \in [0, 1]} I_{\lambda, \mu}^-(\gamma(t)), $$ where $$ \Gamma^- = \{\gamma \in \mathcal{C}([0, 1] : \gamma(0) = 0, \gamma(1) = - t_0 \varphi_1\}. $$ Similar to the proof of Lemma 3.5, we obtain the estimate $$ c_{\lambda, \mu}^- \le \max _{s \in [0, 1]} I_{\lambda, \mu}^- (-st_0 \varphi_1) \le \frac{t_0^q \mu}{q} \|\varphi_1\|_q^q + \frac{t_0^r \lambda}{r} \int_{\Omega} \varphi_1^r dx, $$ which implies that if $\lambda, \mu$ are small enough, then we obtain the estimate $c_{\lambda, \mu}^- < \frac{1}{N} S^{N/2} b^{\frac{2-N}{2}}$, consequently, by the Mountain Pass Theorem, $c_{\lambda, \mu}^-$ is a critical value of $I_{\lambda, \mu}^-$. Hence, we obtain another important result. \begin{lemma}\label{lem3.7} Let $N > 2$, $1 < \min \{q, r\} \le \max \{q, r\} < 2 < \theta \le 2^*$ and $\lambda_1 < a$. If $\lambda, \mu$ small enough, then \eqref{e1.1} has at least a nontrivial negative solution. \end{lemma} For $W_k$ and $V_k^m$ are as in Section 2, we now consider the existence of the third solution. \begin{lemma}\label{lem3.8} There exist $\alpha > 0$ and $\rho > 0$ such that $$ I_{\lambda, \mu}(u) \ge \alpha $$ whenever $u \in W_k$ and $\|u\| = \rho$. \end{lemma} \begin{proof} If $u \in W_k$, then \begin{align*} I_{\lambda, \mu}(u)& = \frac{1}{2} \|u\|^2 + \frac{\mu}{q} \|u\|_q^q + \frac{\lambda}{r} \int_{\Omega} |u|^r dx - \frac{a}{2} \int_{\Omega} |u|^2 dx - \frac{b}{\theta} \int_{\Omega} (u^+)^{\theta} dx\\ &\ge \frac{1}{2} \|u\|^2 - \frac{a}{2} \int_{\Omega} |u|^2 dx - \frac{b}{\theta} \int_{\Omega} (u^+)^{\theta} dx \\ &\ge \big(\frac{1}{2} - \frac{a}{2 \lambda_{k+1}}\big) \|u\|^2 - \frac{b}{\theta} |u|_{\theta}^{\theta} \\&\ge \|u\|^2 \left(A - B \|u\|^{\theta-2}\right), \end{align*} with $A, B > 0$. Then it suffices to take $\rho < (A/B)^{\frac{1}{\theta-2}}$. \end{proof} \begin{lemma} \label{lem3.9} Given $\lambda_0 > 0$ and $\mu_0> 0$, there exist $m_0 \in \mathbb{N}$ and $R > \rho$ such that $$ I_{\lambda, \mu}(u) \le \frac{\mu}{q} \|u\|_q^q + \frac{\lambda}{r} \int_{\Omega} |u|^r dx, $$ whenever $u \in \partial Q_m$, where $Q_m = (B_R \cap V_k^m) \oplus [0, R \varphi_{k+1}^m]$, $m \ge m_0$, $\lambda \le \lambda_0$ and $\mu \le \mu_0$. Henceforth $\partial$ means the boundary relative to subspace $V_k^m$. \end{lemma} \begin{proof} Let $m$ be large enough and $a_k < a$ such that \begin{equation} \lambda_k + c_k m^{2-N} \le a_k < a. \label{e3.3} \end{equation} For $u \in V_k^m$, by Lemma \ref{lem2.1} and \eqref{e3.3} one can obtain \begin{equation} \begin{aligned} I_{\lambda, \mu}(u) &= \frac{1}{2} \|u\|^2 + \frac{\mu}{q} \|u\|_q^q + \frac{\lambda}{r} \int_{\Omega} |u|^r dx - \frac{a}{2} \int_{\Omega} |u|^2 dx - \frac{b}{\theta} \int_{\Omega} (u^+)^{\theta} dx \\ & \le \big(\frac{1}{2} - \frac{a}{2 a_k }\big) \|u\|^2 + \frac{\mu}{q} \|u\|_q^q + \frac{\lambda}{r} \int_{\Omega} |u|^r dx - \frac{b}{\theta} \int_{\Omega} (u^+)^{\theta} dx \\& \le \frac{\mu}{q} \|u\|_q^q + \frac{\lambda}{r} \int_{\Omega} |u|^r dx,\label{e3.4} \end{aligned} \end{equation} and \begin{equation} \begin{aligned} &I_{\lambda, \mu}(\xi \varphi_{k+1}^m)\\ &= \frac{\xi^2}{2} \|\varphi_{k+1}^m\|^2 + \frac{\mu \xi^q}{q} \|\varphi_{k+1}^m\|_q^q + \frac{\lambda \xi^r}{r} \int_{\Omega} |\varphi_{k+1}^m|^r dx \\ &\quad - \frac{a \xi^2}{2} \int_{\Omega} |\varphi_{k+1}^m|^2 dx - \frac{b \xi^{\theta}}{\theta} \int_{\Omega} ((\varphi_{k+1}^m)^+)^{\theta} dx \\&\le \frac{\xi^2}{2} \|\varphi_{k+1}^m\|^2 + \frac{\mu_0 \xi^q}{q} \|\varphi_{k+1}^m\|_q^q + \frac{\lambda_0 \xi^r}{r} \int_{\Omega} |\varphi_{k+1}^m|^r dx - \frac{b \xi^{\theta}}{\theta} \int_{\Omega} ((\varphi_{k+1}^m)^+)^{\theta} dx. \end{aligned} \label{e3.5} \end{equation} Since $\varphi_{k+1}^m \to \varphi_{k+1}$ in $W_0^{1, 2}(\Omega)$ as $m \to \infty$, $\varphi_{k+1}$ changes of sign, and $\theta > 2, q, r$, there exist $m_0 \in \mathbb{N}$ and $R > 0$ such that \begin{equation}I_{\lambda, \mu}(R \varphi_{k+1}^m) \le 0 \quad \forall m \ge m_0.\label{e3.6} \end{equation} Then combining \eqref{e2.2}, \eqref{e3.4} and \eqref{e3.6} leads to \begin{equation} I_{\lambda, \mu}(u) \le \frac{\mu}{q} \|u\|_q^q + \frac{\lambda}{r} \int_{\Omega} |u|^r dx,\label{e3.7} \end{equation} whenever $u \in V_k^m \cup (V_k^m \oplus R \varphi_{k+1}^m)$. By \eqref{e3.5}, there exists $\beta > 0$ satisfying \begin{equation} I_{\lambda, \mu}(\xi \varphi_{k+1}^m) \le \beta, \label{e3.8} \end{equation} for all $\xi \ge 0$ and $m \ge m_0$. Since $a > \lambda_k$, we may take $R > 0$ such that \begin{equation} \begin{aligned} I_{\lambda, \mu}(u) &\le \big(\frac{1}{2} - \frac{a}{2 \lambda_k }\big) \|u\|^2 + \frac{\mu}{q} \|u\|_q^q + \frac{\lambda}{r} \int_{\Omega} |u|^r dx\\ &\le - \beta + \frac{\mu}{q} \|u\|_q^q + \frac{\lambda}{r} \int_{\Omega} |u|^r dx. \end{aligned} \label{e3.9} \end{equation} Hence, by \eqref{e2.2}, \eqref{e3.8} and \eqref{e3.9} we obtain \begin{equation} I_{\lambda, \mu}(u + \xi \varphi_{k+1}^m) = I_{\lambda, \mu}(u) + I_{\lambda, \mu}(\xi \varphi_{k+1}^m) \le \frac{\mu}{q} \|u\|_q^q + \frac{\lambda}{r} \int_{\Omega} |u|^r dx\label{e3.10} \end{equation} for all $m \ge m_0$ and $u \in \partial (B_R \cap V_k^m)$. Thus, by \eqref{e3.7} and \eqref{e3.10}, we complete the proof. \end{proof} \begin{proof}[Proof of Theorem \ref{thm1.1}] For the subcritical case, if $\theta < 2^*$, $\alpha$ is given by Lemma \ref{lem3.8}. Take $\lambda$ and $\mu$ small enough in order that $$ \frac{\mu}{q} \|u\|_q^q + \frac{\lambda}{r} \int_{\Omega} |u|^r dx < \alpha $$ for all $u \in \partial Q_m$. Then by Lemma 3.9 we have $$ I_{\lambda, \mu}(u) < \alpha $$ whenever $u \in \partial Q_m$ and $m \ge m_0$. Applying the Linking Theorem, $I_{\lambda, \mu}$ possesses a critical point $u$ at level $c_{\lambda, \mu}$, where \begin{gather*} c_{\lambda, \mu} = \inf _{\Gamma} \max _{u \in Q_m} I_{\lambda, \mu}(\eta(u)) ,\\ \Gamma = \{\eta \in \mathcal{C}(\overline{Q_m}, W_0^{1, p}(\Omega)); \eta = Id \quad\text{on } \partial Q_m\}, \end{gather*} Finally, since $c_{\lambda, \mu} \ge \alpha$, $I_{\lambda, \mu}(u) \ge \alpha > 0$ and $c_{\lambda, \mu}^{\pm} \to 0$ as $\lambda, \mu \to 0$. Hence, if $\lambda, \mu$ are small enough $c_{\lambda, \mu}^{\pm} < \alpha \le c_{\lambda, \mu}$, and we know that $u$ may be neither of the critical points found above for $I_{\lambda, \mu}^+$ and $I_{\lambda, \mu}^-$; that is, $u$ is the third solution of \eqref{e1.1}. Thus, combining Lemmas \ref{lem3.4} and \ref{lem3.7}, we conclude that \eqref{e1.1} has at least three nontrivial solutions. \end{proof} \begin{proof}[Proof of Theorem \ref{thm1.2}] For the critical case, $\theta = 2^*$. Consider the family of functions taken from \cite{a1}: $$ u_{\epsilon} = \frac{C_N \epsilon^{(N-2)/2}}{(\epsilon^2 + |x|^2)^{(N-2)/2}}, \quad \epsilon > 0, $$ where $$ C_N = (N (N-2))^{(N-2)/4}. $$ Let $u_{\epsilon}^m = \eta u_{\epsilon}$, where $\eta$ is given as section 2, and $Q_m^{\epsilon} = (B_R \cap V_k^m) \oplus [0, R u_{\epsilon}^m]$. Replacing $u_{\epsilon}^m$ by $\varphi_{k+1}^m$ in Lemma \ref{lem3.7}, we obtain $$ I_{\lambda, \mu}(u) \le \frac{\mu}{q} \|u\|_q^q + \frac{\lambda}{r} \int_{\Omega} |u|^r dx, \quad \forall u \in \partial Q_m^{\epsilon} $$ whenever $m$ is large. Hence, to conclude the proof of Theorem \ref{thm1.2}, it remains to show that \begin{equation} \sup _{u \in Q_m^{\epsilon}} I_{\lambda, \mu}(u) < \frac{1}{N} S^{N/2} b^{\frac{2-N}{2}}\label{e3.11} \end{equation} for all $\epsilon$, $\lambda$ and $\mu$ small enough. Let $$ J(u) = \frac{1}{2} \|u\|^2 - \frac{a}{2} \int_{\Omega} |u|^2 dx - \frac{b}{2^*} \int_{\Omega} (u^+)^{2^*} dx. $$ Then, we have $$ I_{\lambda, \mu}(u) = J(u) + \frac{\lambda}{r} \int_{\Omega} |u|^r dx + \frac{\mu}{q} \|u\|_q^q. $$ It is sufficient to prove that there exist $m_0 > 0$ and $\epsilon_0 > 0$ such that $$ \sup _{u \in Q_m^{\epsilon}} J(u) < \frac{1}{N} S^{N/2} b^{\frac{2-N}{2}} $$ for all $m \ge m_0$ and $\epsilon < \epsilon_0$. It is not difficult to obtain the following expressions \cite{a2}: \begin{gather} \int_{\Omega} |\nabla u_{\epsilon}^m|^2 dx = S^{N/2} + O(\epsilon^{N-2}),\label{e3.12}\\ \int_{\Omega} |u_{\epsilon}^m|^{2^*} dx = S^{N/2} + O(\epsilon^{N}). \label{e3.13} \end{gather} Moreover, we obtain \begin{equation} \begin{aligned} \int_{\Omega} |u_{\epsilon}^m|^2 dx &= \int_{B(0, 1/m)} |u_{\epsilon}|^2 dx + O(\epsilon^{N-2}) \\ & \ge \int_{B(0, \epsilon)} \frac{C_N^2 \epsilon^{N-2}}{[2 \epsilon^2]^{N-2}} + \int_{\epsilon < |x| < 1/m} \frac{C_N^2 \epsilon^{N-2} }{[2 |x|^2]^{N-2} } + O(\epsilon^{N-2})\\ & = \begin{cases} d \epsilon^2 |\ln \epsilon| + O(\epsilon^2), & \text{if } N = 4,\\ d \epsilon^2 + O(\epsilon^{N-2}), & \text{if } N \ge 5, \end{cases} \label{e3.14} \end{aligned} \end{equation} where $d$ is a positive constant. If $N = 4$, according \eqref{e3.12}, \eqref{e3.13} and \eqref{e3.14}, one has \begin{align*} \frac{\|u_{\epsilon}^m\|^2 - a |u_{\epsilon}^m|^2}{|u_{\epsilon}^m|_{2^*}^2 } &\le \frac{S^2 - a d \epsilon^2 |\ln \epsilon| + O(\epsilon^2)}{(S^2 + O(\epsilon^{4}))^{1/2} } \\ &= S - a d \epsilon^2 |\ln \epsilon| S^{- 1} + O(\epsilon^2) < S, \end{align*} for $\epsilon > 0$ sufficiently small. And similarly, if $N \ge 5$, we obtain \begin{align*} \frac{\|u_{\epsilon}^m\|^2 - a |u_{\epsilon}^m|^2}{|u_{\epsilon}^m|_{2^*}^2 } &\le \frac{S^{N/2} - a d \epsilon^2 + O(\epsilon^{N-2}) }{(S^{N/2} + O(\epsilon^{N}))^{2/2^*} } \\ & = S - a d \epsilon^2 S^{(2-N)/2} + O(\epsilon^{N-2}) < S, \end{align*} for $\epsilon > 0$ sufficiently small. Let $u = v + t u_{\epsilon}^m \in Q_m^{\epsilon}$. By simple computation, we obtain \begin{equation} \max _{t \ge 0} J(tu_{\epsilon}^m) = \frac{b^{\frac{2-N}{2}}}{N} \Big(\frac{\|u_{\epsilon}^m\|^2 - a |u_{\epsilon}^m|^2}{|u_{\epsilon}^m|_{2^*}^2 } \Big)^{N/2} < \frac{1}{N} S^{N/2} b^{\frac{2-N}{2}}. \label{e3.15} \end{equation} Fix $m_0 > 0$ such that $\lambda_k + c_k m_0^{2-N} \le \sigma < a$. Then, for $m \ge m_0$, we obtain \begin{equation} \begin{aligned} J(v) &= \frac{1}{2} \|v\|^2 - \frac{a}{2} \int_{\Omega} |v|^2 dx - \frac{b}{2^*} \int_{\Omega} (v^+)^{2^*} dx \\ &\le \frac{1}{2} \|v\|^2 - \frac{a}{2} |v|^2 \le \frac{\sigma}{2} |v|^2 - \frac{a}{2} |v|^2 \le 0. \end{aligned} \label{e3.16} \end{equation} From \eqref{e3.15} and \eqref{e3.16}, we obtain $$ J(u) = J(v + t u_{\epsilon}^m) = J(v) + J(t u_{\epsilon}^m) \le J(t u_{\epsilon}^m) < \frac{1}{N} S^{N/2} b^{\frac{2-N}{2}}. $$ So, \eqref{e3.11} holds. \end{proof} Letting $\mu \to 0$ in Theorem \ref{thm1.1} and Theorem \ref{thm1.2}, we easily show that Theorems \ref{thm1.1} and \ref{thm1.2} extend the main results in Paiva and Presoto \cite{p1}. \subsection*{Acknowledgments} The authors want to thank the anonymous referees for their valuable comments and suggestions. This work is supported by Natural Science Foundation of Jiangsu Province (BK2011407) and Natural Science Foundation of China (11271364 and 10771212). \begin{thebibliography}{00} \bibitem{a1} C. O. Alves, Y. Ding; \emph{ Multiplicity of positive solutions to a $p$-Laplacian equation involving critical nonlinearity}, J. Math. Anal. Appl. 279 (2003), 508-521. \bibitem{a2}J.G. Azorero, I. P. Alonso; \emph{Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term}, Trans. Amer. Math. Soc. 323 (1991), 877-895. \bibitem{b1} H. Brezis, L. Nirenberg; \emph{$H^1$ Versus $C^1$ local minimizers}, C. R. Acad. Sci., Paris 1317 (1993), 465-472. \bibitem{c1} M. Calanchi, B. Ruf; \emph{Elliptic equations with one-sided critical growth}, Electron. J. Differential Equations 89 (2002), 21 (electronic). \bibitem{c2} M. Cuesta, D. G. de Figueiredo, P. N. Srikanth; \emph{On a resonant-superlinear elliptic problem}, Calc. Var. Partial Differential Equations 17 (2003), 221-233. \bibitem{f1} D. G. De Figueiredo, J. Yang; \emph{Critical superlinear Ambrosetti-Prodi problems}, Topol. Methods Nonlinear Anal. 14 (1999), 59-80. \bibitem{g1} F. Gazzola, B. Ruf; \emph{Lower-order perturbations of critical growth nonlinearities in semilinear elliptic equations}, Adv. Differential Equations 2 (1997), 555-572. \bibitem{h1} C. He, G. Li; \emph{ The existence of nontrivial solution to the $p$-$q$-Laplacian problem with nonlinearlity asymptotic to $u^{p-1}$ at infinity in $\mathbb{R}^N$ }, Nonlinear Anal. 68 (2008), 1100-1119. \bibitem{l1} G. B. Li, G. Zhang; \emph{ Multiple solutions for the $p$-$q$-Laplacian problem with critical exponent}, Acta Math. Sci. Ser. B 29 (4) (2009), 903-918. \bibitem{m1} S. A. Maranoa, N. S. Papageorgiou; \emph{Constant-sign and nodal solutions of coercive ($p$, $q$)-Laplacian problems}, Nonlinear Anal. 77 (2013), 118-129. \bibitem{m2} D. C. de Morais Filho, F. R. Pereira; \emph{Critical Ambrosetti-Prodi type problems for systems of elliptic equations}, Nonlinear Anal. 68 (2008), 194-207. \bibitem{p1} F. O. de Paiva, A. E. Presoto; \emph{Semilinear elliptic problems with asymmetric nonlinearities}, J. Math. Anal. Appl. 409 (2014), 254-262. \bibitem{p2} N. S. Papageorgiou, V. D. R\v{a}dulescu; \emph{Qualitative phenomena for some classes of quasilinear elliptic equations with multiple resonance}, Applied Mathematics Optimization, 69 (2014), 393-430. \bibitem{r1} B. Ribeiro; \emph{The Ambrosetti-Prodi problem for gradient elliptic systems with critical homogeneous nonlinearity}, J. Math. Anal. Appl. 363 (2010) 606-617. \bibitem{s1} M. Sun; \emph{Multiplicity of solutions for a class of the quasilinear elliptic equations at resonance}, J.Math. Anal. Appl. 386 (2012), 661-668. \bibitem{t1} G. Talenti; \emph{Best constant in Sobolev inequality}, Ann. Math. 110 (1976), 353-372. \bibitem{y1} H. Yin, Z. Yang; \emph{A Class of $p$-$q$-Laplacian type equation with concave-convex nonlinearities in bounded domain}, J. Math. Anal. Appl. 382 (2011), 843-855. \bibitem{y2} H. Yin, Z. Yang; \emph{ Multiplicity of positive solutions to a $p$-$q$-Laplacian equation involving critical nonlinearity}, Nonlinear Anal. 75 (2012), 3021-3035. \end{thebibliography} \end{document}