Two Disjoint and Infinite Sets of Solutions for An Elliptic Equation with Critical Hardy-Sobolev-Maz’ya Term and Concave-Convex Nonlinearities

Abstract

Abstract In this paper, we consider the following critical Hardy-Sobolev-Maz’ya problem { − Δ u = | u | 2 ∗ ( t ) − 2 u | y | t + μ | u | q − 2 u in Ω , u = 0 on ∂ Ω , \begin{cases}-\Delta u=\frac{|u|^{2^*(t)-2} u}{|y|^t}+\mu|u|^{q-2} u & \text { in } \Omega, \\ u=0 & \text { on } \partial \Omega,\end{cases} where Ω is an open bounded domain in ℝ N , which contains some points (0,z*), μ > 0 , 1 < q < 2 , 2 ∗ ( t ) = 2 ( N − t ) N − 2 \mu>0,1<q<2,2^*(t)=\frac{2(N-t)}{N-2} , 0 ≤ t < 2, x = (y, z) ∈ ℝ k × ℝ N−k , 2 ≤ k ≤ N. We prove that if N > 2 q + 1 q − 1 + t $N > 2{{q + 1} \over {q - 1}} + t$ , 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 [1] for the case of the critical Hardy-Sobolev-Maz’ya problem.