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.
Citation
Fengshuang Gao. Yuxia Guo. "Multiple solutions for quasilinear equation involving Hardy critical Sobolev exponents." Topol. Methods Nonlinear Anal. 56 (1) 31 - 61, 2020. https://doi.org/10.12775/TMNA.2019.117
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