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

Abstract

Let $1<p<N$, $p^{\ast }=Np/(N-p)$, $0<s<p$, $p^{\ast }(s)=(N-s)p/(N-p)$, and $\Om\in C^{1}$ be a bounded domain in ${\mathbb{R}}^N$ with $0\in\bar{\Om}.$ In this paper, we study the following problem \[ \begin{cases} -Δ_{p}u=μ|u|^{p^{*}-2}u+\frac{|u|^{p^{*}(s)-2}u}{|x|^{s}}+a(x)|u|^{p-2}u, & \text{in }\Om,\\ u=0, & \text{on }\pa\Om, \end{cases} \] where $\mu \ge 0$ 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.

28pages,no figure

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