Schrödinger–Hardy system without the Ambrosetti–Rabinowitz condition on Carnot groups

Abstract

In this paper, we study the following Schrödinger–Hardy system { − Δ G u − μ ψ 2 r ( ξ ) 2 u = F u ( ξ , u , v )   i n   Ω , − Δ G v − ν ψ 2 r ( ξ ) 2 v = F v ( ξ , u , v )   i n   Ω , u = v = 0     o n   ∂ Ω , where Ω is a smooth bounded domain on Carnot groups G , whose homogeneous dimension is Q ≥ 3 , Δ G denotes the sub-Laplacian operator on G , μ and ν are real parameters, r ( ξ ) is the natural gauge associated with fundamental solution of − Δ G on G , ψ is the geometrical function defined as ψ = | ∇ G r | , and ∇ G is the horizontal gradient associated with Δ G . The difficulty is not only the nonlinearities F u and F v without Ambrosetti–Rabinowitz condition, but also the Hardy terms and the structure on Carnot groups. We obtain the existence of nonnegative solution for this system by mountain pass theorem in a new framework.