Infinitely many sign-changing solutions for an elliptic problem with double critical Hardy-Sobolev-Maz'ya terms

Abstract

In this paper, we investigate the following elliptic problem involving double critical Hardy-Sobolev-Maz'ya terms: $$\{\begin{array}{ll}-\Delta u=\mu \frac{|u{|}^{2^{\ast }(t)-2}u}{|y{|}^t}+\frac{|u{|}^{2^{\ast }(s)-2}u}{|y{|}^s}+a(x)u,& \mathrm{in}\Omega ,\\ u=0,& \mathrm{on}\partial \Omega ,\end{array}$$ where $\mu \ge 0$, $a(x)>0$, $2^{\ast }(t)=\frac{2(N-t)}{N-2}$, $2^{\ast }(s)=\frac{2(N-s)}{N-2}$, $0\le t<s<2$, $x=(y,z)\in {\mathbb{R}}^k\times {\mathbb{R}}^{N-k}$, $2\le k<N$, $(0,z^{\ast })\in \bar{\Omega }$ and $\Omega$ is an bounded domain in ${\mathbb{R}}^N$. Applying an abstract theorem in \cite{sz}, we prove that if $N>6+t$ when $\mu >0,$ and $N>6+s$ when $\mu =0,$ and $\Omega$ satisfies some geometric conditions, then the above problem has infinitely many sign-changing solutions. The main tool is to estimate Morse indices of these nodal solution.

11pages

References in corpus (2)

• A Nonlinear Elliptic PDE with Two Sobolev-Hardy Critical Exponents
• Infinitely many solutions for p-Laplacian equation involving double critical terms and boundary geometry