Multiplicity of solutions for quasilinear elliptic systems with singularity

Abstract

In this paper, we study the existence of multiple solutions for the following quasilinear elliptic system: $$\left\{ \begin{gathered} - \Delta _p u - \mu _1 \frac{{|u|^{p - 2} u}} {{|x|^p }} = \alpha _1 \frac{{u^{p*(t) - 2} }} {{|x|^t }}u + \beta _1 |v|^{\beta _2 } |u|^{\beta _1 - 2_u } ,x \in \Omega , \hfill \\ - \Delta _q v - \mu _2 \frac{{|v|^{q - 2} v}} {{|x|^q }} = \alpha _2 \frac{{v^{q*(s) - 2} }} {{|x|^s }}v + \beta _2 |u|^{\beta _1 } |v|^{\beta _2 - 2_u } ,x \in \Omega , \hfill \\ u(x) = v(x) = 0, \hfill \\ \end{gathered} \right. $$ Multiplicity of solutions for the quasilinear problem is obtained via variational method.