Infinitely many solutions for a elliptic system involving critical Sobolev growth, Hardy potential and concave-convex nonlinearity.

Abstract

In this paper, we will prove the existence of two disjoint and infinite sets of solutions for the following elliptic system with critical Sobolev exponents and Hardy potential
$$
\begin{cases}-\Delta u-t \frac{u}{\vert x\vert^{2}}=\frac{2 \alpha}{\alpha+\beta}\vert u\vert^{\alpha-2} u \vert v\vert^{\beta}+\frac{2 p}{p+q}\vert u\vert^{p-2} u\vert v\vert^{q} & \text { in } \Omega, \\ -\Delta v-t \frac{v}{\vert x\vert^{2}}=\frac{2 \beta}{\alpha+\beta}\vert u\vert^{\alpha}\vert v\vert^{\beta-2} v+\frac{2 q}{p+q}\vert u\vert^{p}\vert v\vert^{q-2} v & \text { in } \Omega, \\
u=v=0 & \text { on } \partial \Omega,\end{cases}
$$
where $\Omega\subset \mathbb{R}^{N}$ is a smoothly bounded domain containing the origin, $N \geq 7,$ \; $\alpha+\beta= 2^{*}$, $\bar{t}=\frac{(N-2)^{2} }{4}$, $ t \in[ 0,\bar{t}-4)$, $2^{*}-\sqrt{1-\frac{t}{\bar{t}}}<p+q<2$ and $2^{*}:=\frac{2 N}{N-2}$ denotes the critical Sobolev exponent.