Infinitely many positive energy solutions for an elliptic equation 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 equation with critical Sobolev exponents and Hardy potential

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u-\frac{\mu }{\vert x\vert ^{2}} u= \vert u\vert ^{2^{*}-2} u + a\vert u\vert ^{q-2} u &{} \; \text {in} \; \Omega , \\ u=0 &{} \; \text {on} \; \partial \Omega , \end{array}\right. \end{aligned}$$

where $\Omega \subset {\mathbb {R}}^{N}$ is a smoothly bounded domain containing the origin, $N \ge 7$, ${\bar{\mu }}=\frac{(N-2)^{2} }{4}$, $\mu \in [ 0,{\bar{\mu }}-4)$, $a>0$, $2^{*}-\sqrt{1-\frac{\mu }{{\bar{\mu }}}}<q<2$ and $2^{*}:=\frac{2 N}{N-2}$ denotes the critical Sobolev exponent.