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

In this paper, we consider the following critical Hardy-Sobolev-Maz’ya problem {−Δu=|u|2∗(t)−2u|y|t+μ|u|q−2u 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&#x003C;q&#x003C;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>2q+1q−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.