Infinitely many solutions for p-Laplacian equation involving double critical terms and boundary geometry

Abstract

Let $1<p<N$, $p^{*}=Np/(N-p)$, $0<s<p$, $p^{*}(s)=(N-s)p/(N-p)$, and $\Om\in C^{1}$ be a bounded domain in $\R^{N}$ with $0\in\bar{\Om}.$ In this paper, we study the following problem \[ \begin{cases} -\Delta_{p}u=\mu|u|^{p^{*}-2}u+\frac{|u|^{p^{*}(s)-2}u}{|x|^{s}}+a(x)|u|^{p-2}u, & \text{in }\Om,\\ u=0, & \text{on }\pa\Om, \end{cases} \] where $\mu\ge0$ is a constant, $\De_{p}$ is the $p$-Laplacian operator and $a\in C^{1}(\bar{\Om})$. By an approximation argument, we prove that if $N>p^{2}+p,a(0)>0$ and $\Omega$ satisfies some geometry conditions if $0\in\partial\Omega$, say, all the principle curvatures of $\partial\Omega$ at $0$ are negative, then the above problem has infinitely many solutions.