Multiple solutions for quasilinear equation involving Hardy critical Sobolev exponents

Abstract

We consider the following quasilinear elliptic equation with critical Sobolev and Hardy-Sobolev exponents: \[ \begin{cases} \displaystyle -\sum\limits_{i,j=1}^ND_j(b_{ij}(v)D_iv)+\frac{1}{2}\sum\limits_{i,j=1}^N b_{ij}'(v)D_ivD_jv \\ \displaystyle \qquad\quad =\frac{|v|^{2^*_sq-2}v}{|x|^s}+\mu|v|^{2^*q-2}v+a(x)|v|^{2q-2}v &\hbox{in }\Omega,\\ v=0 &\hbox{on } \partial\Omega, \end{cases} \] where $b_{ij}\in C^1(\mathbb{R},\mathbb{R})$ satisfies the growth condition $|b_{ij}(t)|\sim|t|^{2(q-1)}$ at infinity, $q\geq1$, $\mu\geq0$, $0< s< 2$, $2^*_s={2(N-s)}/({N-2})$, $2^*={2N}/({N-2})$, $0\in\overline{\Omega}$ and $\Omega$ is a bounded domain in $\mathbb{R}^N$. In this paper, we will investigate the effects of the lower order terms $a(x)|v|^{2q-2}v$ and the growth of $b_{ij}(v)$ at infinity on the existence of infinitely many solutions for the above equations.