Two disjoint and infinite sets of solutions for an elliptic equation with critical Hardy-Sobolev-Maz'ya term and concave-convex nonlinearities

Abstract

In this paper, we consider the following critical Hardy-Sobolev-Maz’ya problem$$ \left\{\begin{array}{ll} -\Delta u= \frac{\vert u\vert^{2^{*}(t)-2} u}{\vert y \vert^{t}} + \mu \vert u \vert ^{q-2} u & \; in \; \Omega, \\ u=0 & \; on \; \partial \Omega, \end{array}\right.$$where $\Omega$ is an open bounded domain in $\mathbb{R}^{N}$, which contains some points $\left(0, z^{*}\right)$, $\mu > 0,$ $ 1<q<2,$ $ 2^{*}(t)=\frac{2(N-t)}{N-2},$ $ 0 \leq t<2,$ $x=(y, z) \in \mathbb{R}^{k} \times \mathbb{R}^{N-k},$ $ 2 \leq k<N$. We prove that if $N>2\frac{q+1}{q-1}+t$, then the above problem has two disjoint and infinite sets of solutions. Here, we give a positive answer to one open problem proposed by Ambrosetti, Brezis and Cerami in \cite{AB} for the case of the critical Hardy-Sobolev-Maz’ya problem.