\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2016 (2016), No. 65, pp. 1--11.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2016 Texas State University.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2016/65\hfil Existence and multiplicity of solutions] {Existence and multiplicity of solutions for elliptic equations with singular growth} \author[Y. Nasri, A. Rimouche \hfil EJDE-2016/65\hfilneg] {Yasmina Nasri, Ali Rimouche} \address{Yasmina Nasri \newline Laboratoire Syst\`emes Dynamiques et Applications \\ Facult\'e des Sciences \\ Universit\'e de Tlemcen BP 119 Tlemcen 13000, Alg\'erie} \email{y\_nasri@hotmail.com} \address{Ali Rimouche \newline Laboratoire Syst\`emes Dynamiques et Applications \\ Facult\'e des Sciences \\ Universit\'e de Tlemcen BP 119 Tlemcen 13000 - Alg\'erie} \email{ali.rimouche@mail.univ-tlemcen.dz} \thanks{Submitted June 6, 2015. Published March 10, 2016.} \subjclass[2010]{35J65, 35J20} \keywords{Semilinear elliptic equation; Hardy potential; \hfill\break\indent critical Sobolev exponent} \begin{abstract} In this article, we consider an elliptic problem with singular and critical growth. We prove the existence and multiplicity of solutions for the resonant and nonresonant cases. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} In this article we study the existence of nontrivial solutions to the semilinear elliptic problem \begin{equation} \begin{gathered} -\Delta u-\mu \frac{u}{|x|^2}=\lambda f(x)u+|u|^{2^{*}-2}u \quad \text{in } \Omega\setminus \{0\}, \\ u=0 \quad \text{on } \partial\Omega, \end{gathered} \label{eq:P-lambda} \end{equation} where $\Omega$ is a bounded domain in $\mathbb{R}^N$ $(N\geq3)$ with $0\in\Omega$, $\lambda$ and $\mu$ are positive parameters such that $0\leq\mu< \bar{\mu}=(\frac{N-2}{2})^2$, $\bar{\mu}$ is the best constant in the Hardy inequality, $2^*=\frac{2N}{N-2}$ is the critical Sobolev exponent and $f$ is a positive singular function which will be specified later. The study of this type of problems is motivated by its various applications. For example, it has been introduced as a model for nonlinear Schr\"odinger equations with a singular potential of the form: \[ -i\hbar\frac{\partial\psi}{\partial t}-\frac{\hbar^2}{2}\Delta\psi+V(x)\psi= |\psi|^{p-1}\psi,\quad(x,t)\in\mathbb{R}^N\times \mathbb{R}^+, \] where $i$ is the imaginary unit and $\hbar$ denotes the Plank constant. This equation describes Bose-Einstein condensates \cite{Lieb,Meystre} and the propagation of light in some nonlinear optical materials \cite{Mills}. Equation \eqref{eq:P-lambda} is doubly critical due to the presence of the critical exponent and the Hardy potential. If $\lambda\leq0$ and $\Omega$ is starshaped, using Pohozaev identity \cite{Pohozaev} one sees that \eqref{eq:P-lambda} has no nontrivial solution. When $f\equiv 1$ the problem \eqref{eq:P-lambda} has been widely investigated, see \cite{Cao,Chen,Ferrero,Jannelli} and the references therein. In Jannelli \cite{Jannelli}, for $f \equiv 1$, the following was proved: (1) If $0\leq\mu\leq\bar{\mu}-1$, then \eqref{eq:P-lambda} has at least one solution $u\in H_0^1(\Omega)$ for all $0<\lambda<\lambda_1^{\mu}$ where $\lambda_1^{\mu}$ is the first eigenvalue of the operator $(-\Delta-\frac{\mu}{|x|^2})$ in $H_0^1(\Omega)$. (2) If $\bar{\mu}-1<\mu<\bar{\mu}$, then \eqref{eq:P-lambda} has at least one solution $u\in H_0^1(\Omega)$ for all $\mu^*<\lambda<\lambda_1^{\mu}$ where \begin{equation*} \mu^{*}={\min_{\varphi\in H_0^1(\Omega)}\frac{\int_{\Omega}{\frac{ |\nabla \varphi(x)|^2}{|x|^{2\gamma}}dx}}{\int_{\Omega} {\frac{|\varphi(x)|^2}{|x|^{2\gamma}}}dx}}, \end{equation*} and $\gamma=\sqrt{\bar{\mu}}+\sqrt{\bar{\mu}-\mu}$. Ferrero and Gazzola \cite{Ferrero} showed the existence of solutions for $\lambda\geq\lambda_1^{\mu}$; Cao and Han \cite{Cao} extended the results in \cite{Ferrero}. When $f\not\equiv 1$, is a positive measurable function, Nasri \cite{Nasri} extended the results of Jannelli \cite{Jannelli} allowing $f$ to be singular. Borrowing ideas from \cite{Cao} and \cite{Ferrero}, we give existence and multiplicity results when $f$ is a singular function. Resonant and non resonant cases are considered. This article is organized as follows: in Section 2 we collect preliminary results and state our main results, in Section 3 we present variational properties of \eqref{eq:P-lambda}, and in Section 4 we complete the proofs of the main results. \section{Preliminaries and statement of main results} Throughout this article we denote by $C,C_1,C_2,\dots$ generic positive constants; $B_R$ is the ball centered at $0$ with radius $R$; $H^{-1}$ is the topological dual of $H_0^1(\Omega)$; $L^p(\Omega)$ for $1\leq p\leq+\infty$, denotes the Lebesgue space with $|\cdot|_p$, its usual norm. For all $0\leq\mu<\bar{\mu}$, we endow the Hilbert space $H_0^1(\Omega):=H_{\mu}(\Omega)$ with the scalar product \begin{equation*} \langle u,\,v\rangle_{\mu}=\int_{\Omega}{\Big(\nabla u\nabla v-\mu\frac{uv}{ |x|^2}Big)dx},\quad \forall u,v\in H_{\mu}(\Omega), \end{equation*} and define \begin{equation*} \|u\|_{\mu}:=\Big(\int_{\Omega}{\big(|\nabla u|^2-\mu \frac{u^2}{|x|^2}\big)dx}\Big)^{1/2},\quad \forall u\in H_{\mu}(\Omega). \end{equation*} By Hardy's inequality \cite{Hardy}, this norm is equivalent to the usual norm in $H^1_0(\Omega)$. Let \begin{equation*} \mathcal{F}_2=\big\{f:\Omega\to\mathbb{R}^{+}:\underset{ |x|\to 0}{\lim}|x|^2f(x)=0\text{ with } f\in L^{\infty}_{\rm loc}(\Omega\setminus\{0\})\big\}. \end{equation*} Next we state several properties to be used later in this paper. \begin{lemma}[\cite{Chaudhuri}] \label{lem:eigenvalue} Let $0\leq\mu<\bar{\mu}$, $\lambda\in\mathbb{R}$, $f\in\mathcal{F}_2$. The eigenvalue problem \begin{equation} \begin{gathered} -\Delta e-\mu\frac{e}{|x|^2}=\lambda f(x)e \quad\text{in }\Omega \\ e=0 \quad\text{on }\partial\Omega, \end{gathered} \label{eq:val-pro} \end{equation} admits non-trivial weak solutions in $H_0^1(\Omega)$, corresponding to \[ \lambda\in\sigma_{\mu}(f):=\big(\lambda_k^{\mu}(f)\big) _{k=1}^{\infty} \] where \begin{equation*} 0<\lambda_1^{\mu}(f)\leq\lambda_2^{\mu}(f)\leq\dots \to+\infty. \end{equation*} if $\Omega$ is $C^{1,1}$, then all weak solutions of \eqref{eq:val-pro} are in $H^{1}_0(\Omega)\cap W^{2,\,r}(\Omega)$ for all $10$, the problem \eqref{eq:P-lambda} admits at least one solution. \end{theorem} We prove our results using critical point theory. However the energy functional associated to \eqref{eq:P-lambda} does not satisfy $\text{(P.S)}$ because of the lack of compactness of the embedding $H_0^{1}( \Omega ) \hookrightarrow L^{2^{\ast }}( \Omega ) $ and $H_0^{1}( \Omega ) \hookrightarrow L^2( \Omega,\,| x| ^{-2}dx) $, standard arguments are not applicable. We follow Brezis-Nirenberg's arguments in \cite{Brezis} to verify that the energy functional to \eqref{eq:P-lambda} satisfies $\text{(P.S)}_{c}$ condition on a suitable compactness range. Then, by employing the technics introduced in \cite{Cao,Ferrero} we obtain some results on Brezis-Nirenberg type problems for an elliptic equation involving critical growth and singular coefficients. \section{Variationnal characterization} The nontrivial solutions to \eqref{eq:P-lambda} are the non zero critical points of the energy functional \begin{equation} J_{\lambda}(u)=\frac{1}{2}\int_{\Omega}{|\nabla u|^2dx}-\frac{ \mu}{2}\int_{\Omega}{\frac{|u|^2}{|x|^2}dx}-\frac{ \lambda}{2}\int_{\Omega}{f|u|^2dx}-\frac{1}{2^{*}}\int_{\Omega}{ |u|^{2^{*}}dx}. \label{eq:J_lambda} \end{equation} Let \begin{equation*} S_{\mu}:=\underset{u\in H^{1}(\mathbb{R}^{N})\setminus\{0\}}{\inf} \frac{\int_{\mathbb{R}^{N}}{(|\nabla u|^2-\mu\frac{u^2}{ |x|^2})dx}}{(\int_{\mathbb{R}^{N}}{|u|^{2^{*}}dx} )^{2/2^{*}}}. \end{equation*} From \cite{Terracini}, we know that $S_{\mu}$ is achieved by the family of functions \begin{equation*} u^{*}_{\varepsilon}(x)=\frac{C_{\varepsilon}}{\big(\varepsilon^2|x |^{\gamma'/\sqrt{\bar{\mu}}}+|x|^{\gamma/\sqrt{\bar{ \mu}}}\big)^{\sqrt{\bar{\mu}}}} \end{equation*} with \[ C_{\varepsilon}=(\frac{4\varepsilon N(\bar{\mu}-\mu)}{ N-2})^{\sqrt{\bar{\mu}}/{2}}, \quad \gamma=\sqrt{\bar{\mu}}+\sqrt{\bar{\mu}-\mu},\quad \gamma'=\sqrt{\bar{\mu}}-\sqrt{\bar{\mu}-\mu}. \] \begin{lemma} \label{lem:07} Assume that $f\in \mathcal{F}_{2,\,\beta }$ and $\mu <\bar{\mu}$, then $J_{\lambda }$ satisfies the $(\mathrm{PS}) _{c}$ condition for all $c<\frac{1}{N}S_{\mu }^{N/2}$. \end{lemma} The proof of the above lemma is the same that in \cite{Chen}. Fix $k\in \mathbb{N}$ and let \begin{equation*} H^{-}=\operatorname{span}\{e_1,\,e_2,\dots ,e_k\},\quad H^{+}=( H^{-})^{\perp }. \end{equation*} Take always $m\in \mathbb{N}$ large enough, so $B_{1/m}\subset \Omega $ and consider the function $\xi _{m}:\Omega \to \mathbb{R}$ defined by \begin{equation*} \xi _{m}(x):= \begin{cases} 0 & \text{if } x\in B_{1/m}(0), \\ m| x| -1 & \text{if } x\in A_{m}=B_{2/m}(0)\setminus B_{1/m}(0), \\ 1 & \text{if } x\in B_{2/m}(0). \end{cases} \end{equation*} Then, as in \cite{Ferrero}, define the approximate eigenfunctions $e_{i}^{m}:=\xi _{m}e_{i}\ $for all $i\in \mathbb{N}$ and the space $H_{m}^{-}:=\operatorname{span}\{e_{i}^{m}$, $i=1,\dots ,k\}$. For all $\varepsilon >0$, consider the shifted functions \begin{equation*} u_{m}^{\varepsilon }(x)= \begin{cases} u_{\varepsilon }^{\ast }(x)-u_{\varepsilon }^{\ast }( \frac{1}{m} ) & \text{if } x\in B_{1/m}(0)\setminus \{0\}, \\ 0 & \text{if } x\in \Omega \setminus B_{1/m}(0). \end{cases} \end{equation*} \begin{lemma}\label{lem:06} For $f\in \mathcal{F}_{2,\beta }$, $\mu <\bar{\mu}$ and $ i\neq j( i,j=1,2,\ldots,k) $, we have: \begin{itemize} \item[(i)] $\|e_{i}^{m}-e_{i}\|_{\mu }\to 0$ as $m\to \infty$, \begin{gather} \|e_k^{m}\|_{\mu }\leq \lambda _k^{\mu }(f)+Cm^{-2\sqrt{\bar{\mu} -\mu }}, \label{eq:16} \\ | \langle e_{i}^{m},\,e_{j}^{m}\rangle _{\mu }| \leq Cm^{-2\sqrt{\bar{\mu}-\mu }}, \label{eq:18} \\ \|e_k^{m}\|_{L_{(\Omega ,f)}^2}\leq \lambda _k^{\mu }(f)+Cm^{-2+\beta \sqrt{\bar{\mu}-\mu }}, \label{eq:19} \end{gather} \item[(ii)] For $\Lambda =\{u\in H_{m}^{-}:\|u\|_{L^2( \Omega,f) }=1\}$, we have \begin{equation*} \max_{u\in \Lambda } \|u\|_{\mu }\leq \lambda _k^{\mu }(f)+Cm^{-2\sqrt{\bar{\mu}-\mu }}. \end{equation*} \end{itemize} \end{lemma} The proof of the above lemma is essentially given in \cite{Cao} with minor modifications. \begin{lemma} Let $0\leq \beta <2$ and $f\in\mathcal{F}_{2,\beta}$. For $m$ large enough and $\varepsilon $ small enough, we have \begin{gather} \int_{\Omega }{\Big( | \nabla u_{m}^{\varepsilon }| -\mu \frac{( u_{m}^{\varepsilon }) ^2}{| x| ^2}\Big) dx}\leq S_{\mu }^{N/2}+C\varepsilon ^{N-2}m^{2 \sqrt{\bar{\mu}-\mu }}, \label{eq:31} \\ \int_{\Omega }{( u_{m}^{\varepsilon }) ^{2^{\ast }}dx} \geq S_{\mu }^{N/2}-C\varepsilon ^{N}m^{2N\sqrt{\bar{\mu}-\mu }/(N-2)}. \label{eq:32} \\ \begin{aligned} &\int_{\Omega }f( u_{m}^{\varepsilon }) ^2dx\\ &\geq \begin{cases} C_1\varepsilon ^{\frac{\sqrt{\bar{\mu}}}{2\sqrt{\bar{\mu}-\mu }}( 2-\beta ) }-C\varepsilon ^{2\sqrt{\bar{\mu}}}m^{-2+\beta +2\sqrt{\bar{ \mu}-\mu }} & \text{if }\mu <\bar{\mu}-( \frac{2-\beta }{2}) ^2. \\ C_2\varepsilon ^{(N-2)/2}| \ln \varepsilon | -C\varepsilon ^{N-2} & \text{if }\mu =\bar{\mu}-( \frac{2-\beta }{2}) ^2. \end{cases} \end{aligned} \label{eq:32b} \end{gather} \end{lemma} \begin{proof} For the proof of \eqref{eq:31} and \eqref{eq:32} we argue as in \cite{Ferrero}. We prove only \eqref{eq:32b}. Since $f\in \mathcal{F}_{2,\,\beta }$, we have \begin{equation*} \int_{\Omega }{f( u_{\varepsilon }^{\ast }) ^2dx} \geq \begin{cases} C_1\varepsilon ^{\sqrt{\bar{\mu}}(2-\beta )/2\sqrt{\bar{\mu}-\mu }} & \text{if } \mu <\bar{\mu}-( \frac{2-\beta }{2}) ^2 \\ C_2\varepsilon ^{(N-2)/2}| \ln \varepsilon | & \text{if } \mu =\bar{\mu}-( \frac{2-\beta }{2}) ^2 \end{cases} \end{equation*} and \begin{align*} \int_{\Omega }f\big( u_{m}^{\varepsilon }\big) ^2dx &=\int_{\Omega }f( u_{\varepsilon }^{\ast }(x) -u_{\varepsilon }^{\ast }( \frac{1}{m}) ) ^2dx \\ &\geq \int_{\Omega }f( u_{\varepsilon }^{\ast }) ^2dx-2\int_{\Omega }f\frac{u_{\varepsilon }^{\ast }C_{\varepsilon }}{ \big( \varepsilon ^2( \frac{1}{m}) ^{\gamma '/\sqrt{ \bar{\mu}}}+( \frac{1}{m}) ^{\gamma /\sqrt{\bar{\mu}}}\big) ^{ \sqrt{\bar{\mu}}}}dx \\ &\geq \int_{\Omega }f( u_{\varepsilon }^{\ast })^2dx\\ &\quad -C\int_{\Omega }f\frac{\varepsilon ^{2\sqrt{\bar{\mu}}}}{\big( \varepsilon ^2|x|^{\gamma '/\sqrt{\bar{\mu}}}+|x|^{\gamma /\sqrt{ \bar{\mu}}}\big) ^{\sqrt{\bar{\mu}}}\big( \varepsilon ^2( \frac{1}{ m}) ^{\gamma '/\sqrt{\bar{\mu}}}+( \frac{1}{m}) ^{\gamma /\sqrt{\bar{\mu}}}\big) ^{\sqrt{\bar{\mu}}}}dx. \end{align*} We have \begin{align*} \frac{\varepsilon ^{2\sqrt{\bar{\mu}}}}{( \varepsilon ^2( \frac{1 }{m}) ^{\gamma '/\sqrt{\bar{\mu}}}+( \frac{1}{m}) ^{\gamma /\sqrt{\bar{\mu}}}) ^{\sqrt{\bar{\mu}}}} &=\frac{\varepsilon ^{2\sqrt{\bar{\mu}}}}{\varepsilon ^{2\sqrt{\bar{\mu}}}( \frac{1}{m} ) ^{\gamma '}( 1+\varepsilon ^{-2}( \frac{1}{m} ) ^{2\sqrt{\bar{\mu}-\mu }/\sqrt{\bar{\mu}}}) ^{\sqrt{\bar{\mu}}} } \\ &\leq \varepsilon ^{2\sqrt{\bar{\mu}}}m^{\gamma }, \end{align*} and \begin{align*} &\int_{B_{1/m}}f\frac{dx}{( \varepsilon ^2|x|^{\gamma '/\sqrt{ \bar{\mu}}}+|x|^{\gamma /\sqrt{\bar{\mu}}}) ^{\sqrt{\bar{\mu}}}} \\ &\leq C\int_0^{1/m}\frac{r^{N-1-\beta }dr}{( \varepsilon ^2r^{\gamma '\sqrt{\bar{\mu}}}+r^{\gamma /\sqrt{\bar{\mu}}}) ^{\sqrt{\bar{ \mu}}}} \\ &\leq C\varepsilon ^{\frac{( N-\beta -\gamma ') \sqrt{ \bar{\mu}}}{\sqrt{\bar{\mu}-\mu }}-2\sqrt{\bar{\mu}}} \int_0^{1/m \varepsilon ^{\frac{\sqrt{\bar{\mu}}}{\sqrt{\bar{\mu}-\mu }}}}\frac{\tau ^{N-1-\beta }d\tau }{\tau ^{\gamma '}\tau ^{\frac{2\sqrt{\bar{\mu} -\mu }}{\sqrt{\bar{\mu}}}\sqrt{\bar{\mu}}}} \\ &\leq Cm^{-\gamma '-2+\beta }. \end{align*} Hence \begin{align*} \int_{\Omega }f( u_{m}^{\varepsilon }) ^2dx &\geq \int_{\Omega }f( u_{\varepsilon }^{\ast }) ^2dx-C\varepsilon ^{2\sqrt{\bar{ \mu}}}m^{\gamma }m^{-\gamma '-2+\beta } \\ &\geq \int_{\Omega }f( u_{\varepsilon }^{\ast }) ^2dx-C\varepsilon ^{2\sqrt{\bar{\mu}}}m^{-2+\beta +2\sqrt{\bar{\mu}-\mu }}. \end{align*} For $\mu \leq \bar{\mu}-( \frac{2-\beta }{2}) ^2$ we find the result. \end{proof} Now, we prove that the functional $J_{\lambda }$ has Linking geometry. \begin{proposition}\label{prop:3.4} Suppose that $f\in \mathcal{F}_{2,\beta }$ and there exists $k\in \mathbb{N}^{\ast }$ such that $\lambda _k^{\mu }(f)\leq \lambda <\lambda _{k+1}^{\mu }(f)$. Then: \begin{itemize} \item[(i)] There exist $\rho , \alpha >0$ such that $J_{\lambda }\mid _{\partial B_{\rho }\cap H^{+}}\geq \alpha $, \item[(ii)] There exists $R>\rho $ such that $J_{\lambda }\mid _{\partial Q_{m}^{\varepsilon }}\leq p(m)$ with $p(m)\to 0$ as $m\to +\infty $ where $Q_{m}^{\varepsilon }=( \overline{B_{R}}\cap H_{m}^{-}) \oplus \{r.u_{m}^{\varepsilon }:00 $ such that \[ J_{\lambda }\mid _{\partial B_{\rho }\cap H^{+}}\geq \alpha . \] For any $u\in H_{m}^{-}$, from the estimates of Lemma \ref{lem:06} we obtain \begin{equation} \begin{aligned} J_{\lambda }( u) &\leq C_1m^{-2\sqrt{\bar{\mu}-\mu } }\int_{\Omega }fu^2dx-\frac{1}{2^{\ast }}\int_{\Omega }u^{2^{\ast }}dx\\ &\leq C_2m^{-2\sqrt{\bar{\mu}-\mu }}| u| _{2^{\ast }}^2-\frac{1}{2^{\ast }}| u| _{2^{\ast }}^{2^{\ast }} \\ &\leq C_{3}m^{-N\sqrt{\bar{\mu}-\mu }}. \end{aligned} \label{eq:3.3} \end{equation} Consequently, \[ \lim_{m\to \infty } \max_{u\in H_{m}^{-}}J_{\lambda }(u)=0. \] On the other hand, \begin{equation*} J_{\lambda }( ru_{m}^{\varepsilon }) \leq \frac{r^2}{2} \| u_{m}^{\varepsilon }\| _{\mu }-\frac{r^{2^{\ast }}}{ 2^{\ast }}| u_{m}^{\varepsilon }| _{2^{\ast }}^{2^{\ast}}; \end{equation*} then $J_{\lambda }( ru_{m}^{\varepsilon }) $ becomes negative if $r=R$ and $R$ large enough. Therefore \begin{equation*} J_{\lambda }(u)\leq Cm^{-N\sqrt{\bar{\mu}-\mu }}\quad \text{for all } u\in H_{m}^{-}\cup \{ H_{m}^{-}\oplus Ru_{m}^{\varepsilon }\} . \end{equation*} Since $\max_{0\leq r\leq R} J_{\lambda }( ru_{m}^{\varepsilon }) <+\infty $ as $v\in H_{m}^{-}\oplus \mathbb{R}^{+}u_{m}^{ \varepsilon }$, we may write $v=u+ru_{m}^{\varepsilon }$ with $|\operatorname{supp}( u_{m}^{\varepsilon }) \cap \operatorname{supp}(u)| =0$, then for large $R$, \begin{equation*} J_{\lambda }\mid _{\partial Q_{m}^{\varepsilon }}\leq 0. \end{equation*} \end{proof} \section{Proof of theorem \ref{thm:01}} \begin{lemma}\label{lem:12} Suppose that $f\in\mathcal{F}_{2,\,\beta}$ and $\mu \in [0,\,\bar{\mu}-( \frac{2-\beta }{2}) ^2] $. Then \begin{equation*} J_{\lambda }( t_{\varepsilon }u_{m}^{\varepsilon }) <\frac{1}{N} S_{\mu }^{N/2}\quad for\ \varepsilon \text{\ small\ enough}. \end{equation*} \end{lemma} \begin{proof} Assume by contradiction that for all $\varepsilon >0$, there exists $t_{\varepsilon }>0$ such that \begin{equation} J_{\lambda }( t_{\varepsilon }u_{m}^{\varepsilon }) \geq \frac{1}{ N}S_{\mu }^{N/2}, \label{eq:33} \end{equation} then we affirm that there exists a subsequence of $( t_{\varepsilon }) $ such that $t_{\varepsilon }\to t_0$. If not suppose that $t_{\varepsilon }\to +\infty $, then $J_{\lambda }(t_{\varepsilon }u_{m}^{\varepsilon }) \to -\infty $ when $\varepsilon \to 0$, which contradicts \eqref{eq:33}, thus $(t_{\varepsilon }) $ is bounded and there exists $t_0\geq 0$ such that $t_{\varepsilon }\to t_0$. If $t_0=0$, using the continuity of the embedding, we obtain that $\int_{\Omega }{fu_{m}^{\varepsilon }dx}$ and $|u_{m}^{\varepsilon }| _{2^{\ast }}$ are bounded, the same for $\| u_{m}^{\varepsilon }\| _{\mu }$. We have \begin{equation*} \frac{t_{\varepsilon }^2}{2}\Big[ \int_{\Omega }| \nabla u_{m}^{\varepsilon }| ^2dx-\frac{\mu }{2}\int_{\Omega }\frac{ ( u_{m}^{\varepsilon }) ^2}{|x|^2}dx-\frac{\lambda t_{\varepsilon }^2}{2}\int_{\Omega }f( u_{m}^{\varepsilon }) ^2dx\Big] -\frac{t_{\varepsilon }^{2^{\ast }}}{2^{\ast }}\int_{\Omega }( u_{m}^{\varepsilon }) ^{2^{\ast }}dx=o(1), \end{equation*} which is in contradiction with \eqref{eq:33}. So $t_{\varepsilon}\to t_0>0$. Using \eqref{eq:31} and \eqref{eq:32} and letting $\varepsilon \to 0$, it follows that \begin{gather*} \frac{1}{2}\| t_{\varepsilon }u_{m}^{\varepsilon }\| _{\mu}^2 \leq \frac{1}{2}S_{\mu }^{N/2}+\frac{t_{\varepsilon }^2-1}{2}S_{\mu }^{N/2}+C\varepsilon ^{N-2}m^{2\sqrt{\bar{\mu}-\mu }}, \\ -\frac{1}{2^{\ast }}| t_{\varepsilon }u_{m}^{\varepsilon }| _{2^{\ast }}^{2^{\ast }} \leq -\frac{1}{2^{\ast }}S_{\mu }^{N/2}-\frac{1}{2^{\ast }}( t_{\varepsilon }^{2^{\ast }}-1) S_{\mu }^{N/2}+C\varepsilon ^{N}m^{2N\sqrt{\bar{\mu}-\mu }/(N-2)}. \end{gather*} By adding these two equations, we obtain \begin{equation*} \frac{1}{2}\| t_{\varepsilon }u_{m}^{\varepsilon }\| _{\mu }^2-\frac{ 1}{2^{\ast }}| t_{\varepsilon }u_{m}^{\varepsilon }| _{2^{\ast }}^{2^{\ast }}\leq \frac{1}{N}S_{\mu }^{N/2}+\frac{1}{2}\Big( t_{\varepsilon }^2-1-\frac{N-2}{N}\big(t_{\varepsilon }^{2^{\ast }}-1\big) \Big)S_{\mu }^{N/2}+C\varepsilon ^{N-2}. \end{equation*} By the fact that $\underset{x\geq 0}{\max }\Big(x^2-1-\frac{N-2}{N} (x^{2^{\ast }}-1)\Big)=0$, we obtain \begin{equation*} \frac{1}{2}\| t_{\varepsilon }^2u_{m}^{\varepsilon }\| _{\mu }^2-\frac{1}{2^{\ast }}\int_{\Omega }( t_{\varepsilon }u_{m}^{\varepsilon }) ^{2^{\ast }}\leq \frac{1}{N}S_{\mu }^{N/2}+C\varepsilon ^{N-2}. \end{equation*} We will estimate $\int_{\Omega }{f( t_{\varepsilon }u_{m}^{\varepsilon }) ^2}$ for $\mu \leq \bar{\mu}-( \frac{2-\beta }{2}) ^2$. For $q=1/2^{1/\gamma '}$, we can take $\varepsilon $ small enough so that \begin{equation*} \varepsilon ^{\sqrt{\bar{\mu}}/\sqrt{\bar{\mu}-\mu }}<\frac{1}{qm}. \end{equation*} Hence there exists $C>0$ such that \begin{equation*} \varepsilon ^2|x|^{\gamma '/\sqrt{\bar{\mu}}}+|x|^{\gamma /\sqrt{ \bar{\mu}}}\leq C|x|^{\gamma /\sqrt{\bar{\mu}}},\quad \forall | x| \geq \varepsilon ^{\sqrt{\bar{\mu}}/\gamma }. \end{equation*} and \begin{align*} \int_{\Omega }f( t_{\varepsilon }u_{m}^{\varepsilon }) ^2 &\geq C\int_{\varepsilon ^{\sqrt{\bar{\mu}}/{\gamma }}}^{1/qm}r^{-\beta }\Big( u_{\varepsilon }^{\ast }(r)-u_{\varepsilon }^{\ast }(\frac{1}{m})\Big) ^2r^{N-1}dr \\ &\geq C\int_{\varepsilon ^{\sqrt{\bar{\mu}}/{\gamma }}}^{1/qm}r^{-\beta }( u_{\varepsilon }^{\ast }(r)) ^2r^{N-1}dr \\ &\geq CC_{\varepsilon }^2\int_{\varepsilon ^{\sqrt{\bar{\mu}}/{\gamma } }}^{1/qm}r^{-\beta }r^{-2\gamma }r^{N-1}dr \\ &\geq CC_{\varepsilon }^2\int_{\varepsilon ^{\sqrt{\bar{\mu}}/{\gamma } }}^{1/qm}r^{-\beta +1-2\sqrt{\bar{\mu}-\mu }}dr. \end{align*} To continue we distinguish two cases:\\ (1) $\mu <\bar{\mu}-( \frac{2-\beta }{2}) ^2$, \begin{align*} \int_{\Omega }f( t_{\varepsilon }u_{m}^{\varepsilon }) ^2dx &\geq C\varepsilon ^{2\sqrt{\bar{\mu}}}\varepsilon ^{2( \sqrt{\bar{\mu} }/\gamma ) ( 2-\beta -2\sqrt{\bar{\mu}-\mu }) } \\ &\geq C\varepsilon ^{N-2}\varepsilon ^{2( \sqrt{\bar{\mu}}/\gamma ) ( 2-\beta -2\sqrt{\bar{\mu}-\mu }) }. \end{align*} (2) $\mu =\bar{\mu}-( \frac{2-\beta }{2}) ^2$ \begin{align*} \int_{\Omega }f( t_{\varepsilon }u_{m}^{\varepsilon }) ^2dx &\geq CC_{\varepsilon }^2\int_{\varepsilon ^{\frac{\sqrt{\bar{\mu}}}{ \gamma }}}^{1/qm}r^{-\beta +1-2\sqrt{\bar{\mu}-\mu }}dr \\ &\geq C\varepsilon ^{2\sqrt{\bar{\mu}}}| \ln \varepsilon ^{2( \sqrt{\bar{\mu}}/\gamma ) }| . \end{align*} Thus $J_{\lambda }( t_{\varepsilon }u_{m}^{\varepsilon }) <\frac{1 }{N}S_{\mu }^{N/2}$ for $\mu \in [ 0,\,\bar{\mu}-( \frac{2-\beta }{ 2}) ^2] $. \end{proof} \begin{proof}[Proof of Theorem \ref{thm:01}] The proof is based on Linking Theorem \cite{AR}. We have \begin{equation} \inf_{h\in \Gamma }\max_{u\in Q_{m}^{\varepsilon }} J_{\lambda }( h( u) ) \leq \max_{u\in Q_{m}^{\varepsilon }} J_{\lambda }( u) . \label{eq:34} \end{equation} It suffices to show that \begin{equation*} \max_{u\in Q_{m}^{\varepsilon }} J_{\lambda }(u)<\frac{1}{N}S_{\mu}^{N/2} \end{equation*} Arguing by contradiction, suppose that \begin{equation*} \max_{u\in Q_{m}^{\varepsilon }} J_{\lambda }(u)\geq \frac{1}{N} S_{\mu }^{N/2}\quad \forall m\in \mathbb{N},\quad \forall \varepsilon >0. \end{equation*} Since $\{v\in Q_{m}^{\varepsilon }:J_{\lambda }(v)\geq 0\}$ is a compact set then the upper bound in \eqref{eq:34} is achieved thus, for all $\varepsilon >0$ there exist $\omega _{\varepsilon }\in H_{m}^{-}$ and $t_{\varepsilon }\geq 0$ such that for $v_{\varepsilon }:=\omega _{\varepsilon }+t_{\varepsilon }u_{m}^{\varepsilon }$, we have \begin{equation*} J_{\lambda }( v_{\varepsilon }) :=\sup_{u\in Q_{m}^{\varepsilon }} J_{\lambda }(u)\geq \frac{1}{N}S_{\mu }^{N/2}, \end{equation*} i.e., \begin{equation} \frac{1}{2}\|v_{\varepsilon }\|_{\mu }^2-\frac{\lambda }{2} \int_{\Omega }{fv_{\varepsilon }^2dx}-\frac{1}{2^{\ast }}\int_{\Omega }{ v_{\varepsilon }^{2^{\ast }}dx}\geq \frac{1}{N}S_{\mu }^{N/2},\quad \forall \varepsilon >0. \label{eq:35} \end{equation} Using the proof of Lemma \ref{lem:12}, we obtain that $(t_{\varepsilon })$ admits a convergent subsequence, $(\omega _{\varepsilon })$ is bounded and thus \begin{equation*} t_{\varepsilon }\to t_0>0,\quad \omega _{\varepsilon }\to \omega _0\in H_{m}^{-}. \end{equation*} By the Lemma \ref{lem:06} and the fact that $\lambda \in \big(\lambda _k^{\mu }(f),\,\lambda _{k+1}^{\mu }(f)\big)$, we obtain \begin{align*} J_{\lambda }( \omega _{\varepsilon }) &=\frac{1}{2}\|\omega _{\varepsilon }\|_{\mu }^2-\frac{\lambda }{2}\int_{\Omega }{f\omega _{\varepsilon }^2dx}-\frac{1}{2^{\ast }}\int_{\Omega }{\omega _{\varepsilon }^{2^{\ast }}dx} \\ &\leq \frac{\lambda _k^{\mu }(f)+o(1)}{2}\big|\omega _{\varepsilon }\big| _2^2-\frac{\lambda }{2}\big|\omega _{\varepsilon }\big|_2^2\leq 0 \end{align*} for $m$ large enough. Using \eqref{eq:35} and proceeding in the same way that Lemma \ref{lem:12}, we obtain \begin{equation*} J_{\lambda }\big(v_{\varepsilon }\big)=J_{\lambda }\big(\omega _{\varepsilon }\big)+J_{\lambda }\big(t_{\varepsilon }u_{m}^{\varepsilon }\big)\leq J_{\lambda }\big(t_{\varepsilon }u_{m}^{\varepsilon }\big)<\frac{1}{N}S_{\mu }^{N/2} \end{equation*} which is absurd. \end{proof} \section{Proof of Theorem \protect\ref{thm:02}} Let \begin{equation*} \lambda_{+}=\min\{\lambda_{j}^{\mu}(f)\in\sigma :\lambda<\lambda_{j}^{\mu}(f)\}, \end{equation*} denote by $M\big(\lambda_{j}^{\mu}(f)\big)$ the eigenspace correpsonding to $ \lambda_{j}^{\mu}(f)$. We put \begin{equation*} M^{+}=\overline{\oplus_{\lambda_{j}^{\mu}(f)\geq\lambda_{+}} M\big(\lambda_{j}^{\mu}(f)\big)} ^{H_{\mu}},\quad M^{-}=\oplus_{ \lambda_{j}^{\mu}(f)\leq\lambda_{+}} M\big(\lambda_{j}^{\mu}(f) \big), \end{equation*} suppose that $\lambda_{+}-\lambda0$ and $\delta_{\lambda}\in\big(0,\beta_{\lambda}\big)$ such that $J_{\lambda}(u)\geq\delta_{\lambda}$ for all $u\in M^{+}$ with $\|u\|_{\mu}=\rho_{\lambda}$ \end{lemma} \begin{proof} For all $u\in M^{-}$ we have $\|u\|_{\mu }^2\leq \lambda _{+}\int_{\Omega }{fu^2dx}$. Since $M^{-}$ is a finite dimension space, using H\"older inequality and knowing that \begin{equation*} \max_{t\geq 0} ( A\frac{t^2}{2}-B\frac{t^{2^{\ast }}}{ 2^{\ast }}) =\frac{1}{N}A( \frac{A}{B}) ^{(N-2)/2}\quad \text{for all }A,B>0, \end{equation*} we obtain \begin{align*} J_{\lambda }(u) &= \frac{1}{2}\int_{\Omega }{\big|\nabla u\big|^2dx} -\frac{\mu }{2}\int_{\Omega }{\frac{u^2}{\big|x\big|^2}dx} -\frac{\lambda }{2}\int_{\Omega }{fu^2dx} -\frac{1}{2^{\ast }}\int_{\Omega }{|u|^{2^{\ast }}dx} \\ &\leq \frac{1}{2}\big(\lambda _{+}-\lambda \big)\int_{\Omega }{fu^2dx} -\frac{1}{2^{\ast }}\int_{\Omega }{|u|^{2^{\ast }}dx} \\ &\leq \frac{1}{2}\big(\lambda _{+}-\lambda \big)\int_{\Omega }{|x|^{-\beta }u^2dx}-\frac{1}{2^{\ast }}\int_{\Omega }|u|^{2^{\ast }}dx \\ &\leq \int_{\Omega }\underset{t\geq 0}{\max }\Big(\frac{1}{2} \big(\lambda_{+}-\lambda \big)|x|^{-\beta }t^2-\frac{1}{2^{\ast }} t^{2^{\ast }}\Big)dx. \end{align*} Let $u\in M^{+}$, by the inequality $\|u\|_{\mu }^2\geq \lambda _{+}\int_{\Omega }{fu^2dx}$ and $\|u\|_{\mu }^2\geq S_{\mu }\big| u\big|_{2^{\ast }}^2$, we have \begin{align*} J_{\lambda }(u) &\geq \frac{\lambda _{+}-\lambda }{2\lambda _{+}}\|u \|_{\mu }^2-\frac{1}{S_{\mu }^{2/2^{\ast }}2^{\ast }}\|u\| _{\mu }^{2^{\ast }} \\ &\geq \max_{t\geq 0} \Big(\frac{\lambda _{+}-\lambda }{2\lambda _{+}}t^2-\frac{1}{S_{\mu }^{2/2^{\ast }}2^{\ast }}t^{2^{\ast }}\Big) \\ &= \frac{1}{N}\Big(\frac{\lambda _{+}-\lambda }{2\lambda _{+}}\Big) ^{N/2}S_{\mu }^{N/2}. \end{align*} If we take \[ \rho _{\lambda }=\Big(\big(\frac{\lambda _{+}-\lambda }{ \lambda _{+}}\big)S_{\mu }^{2/2^{\ast }}\Big)^{(N-2)/4},\quad \delta _{\lambda }<\frac{1}{N}\big(\frac{\lambda _{+}-\lambda }{\lambda _{+}}\big) ^{N/2}S_{\mu }^{N/2}, \] then we obtain $J_{\lambda }(u)\geq \delta _{\lambda} $ for all $u\in M^{+}\cap \partial B_{\rho _{\lambda }}$. It remains to show that $\delta _{\lambda }<\beta _{\lambda }$. Indeed, since $M^{+}\cap M^{-}=M\big(\lambda _{+}\big)$, we have $M^{+}\cap M^{-}\cap B_{\rho _{\lambda }}\neq \emptyset $ and all $u\in M^{+}\cap M^{-}\cap B_{\rho _{\lambda }}$ satisfies $\delta _{\lambda }0$, $c<\frac{1}{N}S_{\mu }^{\frac{N}{2}}$. \end{proposition} \begin{proof} Without loss of generality, we can assume that there exists $k$ such that $\lambda_k^{\mu}(f)\leq\lambda<\lambda_{k+1}^{\mu}(f)$. Let $\max_{u\in Q_{m}^{\varepsilon }} J_{\lambda }( u) =J_{\lambda }( w_{m}+t_{m}^{\varepsilon }u_{m}^{\varepsilon }) $, where $w_{m}\in H_{m}^{-}$. Using the same calculation as in the second point of Proposition \ref{prop:3.4}, we have \begin{equation*} J_{\lambda }( w_{m}) \leq Cm^{-N\sqrt{\bar{\mu}-\mu }}. \end{equation*} By choosing $\varepsilon =m^{-\frac{N+2}{N-2}\sqrt{\bar{\mu}-\mu }}$, \begin{gather*} \int_{\Omega }\big( | \nabla u_{m}^{\varepsilon }| ^2-\mu \frac{( u_{m}^{\varepsilon }) ^2}{|x|^2}\big) dx\leq S_{\mu }^{N/2}+Cm^{-N\sqrt{\bar{\mu}-\mu }}, \\ \int_{\Omega }( u_{m}^{\varepsilon }) ^{2^{\ast }}dx\geq S_{\mu }^{N/2}-Cm^{( -N^2/(N-2)) \sqrt{\bar{\mu}-\mu }}, \\ \int_{\Omega }f( u_{m}^{\varepsilon }) ^2dx\geq Cm^{-(N+2) ( \frac{2-\beta }{2}) }. \end{gather*} and \begin{align*} c &\leq \max_{u\in Q_{m}^{\varepsilon }} J_{\lambda }(u) \\ &\leq J_{\lambda }( w_{m}+t_{m}^{\varepsilon }u_{m}^{\varepsilon}) \\ &\leq J_{\lambda }( w_{m}) +J_{\lambda }( t_{m}^{\varepsilon }u_{m}^{\varepsilon }) \\ &\leq Cm^{-N\sqrt{\bar{\mu}-\mu }}+\frac{( t_{m}^{\varepsilon }) ^2}{2}\int_{\Omega }( | \nabla u_{m}^{\varepsilon }| ^2-\mu \frac{( u_{m}^{\varepsilon }) ^2}{|x|^2} -\lambda \int_{\Omega }f( u_{m}^{\varepsilon }) ^2) dx\\ &\quad -\frac{( t_{m}^{\varepsilon }) ^{2^{\ast }}}{2^{\ast }} \int_{\Omega }( u_{m}^{\varepsilon }) ^{2^{\ast }}dx \\ &\leq Cm^{-N\sqrt{\bar{\mu}-\mu }}+\frac{( t_{m}^{\varepsilon }) ^2}{2}( S_{\mu }^{\frac{N}{2}}+Cm^{-N\sqrt{\bar{\mu}-\mu } }-Cm^{-( N+2) ( \frac{2-\beta }{2}) }) \\ &\quad -\frac{( t_{m}^{\varepsilon }) ^{2^{\ast }}}{2^{\ast }}( S_{\mu }^{ \frac{N}{2}}-Cm^{-\frac{N^2}{N-2}\sqrt{\bar{\mu}-\mu }}) \\ &\leq Cm^{-N\sqrt{\bar{\mu}-\mu }}+\frac{1}{N}( S_{\mu }^{\frac{N}{2} }+Cm^{-N\sqrt{\bar{\mu}-\mu }}-Cm^{-( N+2) ( \frac{2-\beta }{ 2}) }) \\ &\quad\times \Big( \frac{S_{\mu }^{\frac{N}{2}}+Cm^{-N\sqrt{\bar{\mu} -\mu }}-Cm^{-( N+2) ( \frac{2-\beta }{2}) }}{S_{\mu }^{ \frac{N}{2}}-Cm^{-\frac{N^2}{N-2}\sqrt{\bar{\mu}-\mu }}}\Big) ^{\frac{N-2}{2}}\,. \end{align*} Note that for $\mu <\bar{\mu}-( \frac{N+2}{N})^2( \frac{2-\beta }{2}) ^2$, we have $( N+2) ( \frac{2-\beta }{2})