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Topological Methods in Nonlinear Analysis

Multiple solutions for quasilinear equation involving Hardy critical Sobolev exponents
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Multiple solutions for quasilinear equation involving Hardy critical Sobolev exponents

Authors

  • Fengshuang Gao
  • Yuxia Guo https://orcid.org/0000-0003-3798-6275

Keywords

Quasilinear equation, critical Hardy-Sobolev exponent, multiple solutions

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.

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Published

2020-05-30

How to Cite

1.
GAO, Fengshuang and GUO, Yuxia. Multiple solutions for quasilinear equation involving Hardy critical Sobolev exponents. Topological Methods in Nonlinear Analysis. Online. 30 May 2020. Vol. 56, no. 1, pp. 31 - 61. [Accessed 6 July 2025].
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