Abstract:

In this paper, we study the existence of multiple solutions for a class of nonlocal quasilinear elliptic problem $\left\{\begin{array}{ll} M\Big(\int_{\mathbb{R} ^{N}}(|\nabla u|^{p}+V(x)|u|^{p}){\rm d}x\Big)(-\Delta_{p}u+V(x)|u|^{p-2}u)=\sigma d^{-1}F_{u}(x, u, v)+\lambda|u|^{q-2}u, \nonumber\\ M\Big(\int_{\mathbb{R} ^{N}}(|\nabla v|^{p}+V(x)|v|^{p}){\rm d}x\Big)(-\Delta_{p}v+V(x)|v|^{p-2}v)=\sigma d^{-1}F_{v}(x, u, v)+\mu|v|^{q-2}v, \nonumber\\ u, v\in W^{1, p}(\mathbb{R} ^{N}), x\in\mathbb{R} ^{N}\nonumber \end{array}\right. $ where $M(s)=s^{k}, k>0, N\geq3, 1<p<q\leq d<p^{\ast}\leq N, \lambda, \mu>0, \sigma\in\mathbb{R} ^{N}$, and in which $p^{\ast}=\frac{Np}{N-p}, $ and $p^{\ast}=\infty$ if $p=N.$ The weight function$V(x)\in C(\mathbb{R} ^{N})$ satisfy some conditions.

Key words: Elliptic equation, Symmetric mountain pass lemma, $PS$) condition')">($PS$) condition