Resumen de Existence and Asymptotic Behaviour of Solutions for a Quasilinear Schrödinger-Poisson System in R

Chongqing Wei, Anran Li, Leiga Zhao

• In this paper, we study the existence and asymptotic behaviour of solutions for the following quasilinear Schrödinger-Poisson system in R3 −u + V(x)u + λφu = f (x, u), x ∈ R3, −φ − ε44φ = λu2, x ∈ R3, where λ and ε are positive parameters, 4 = div(|∇u| 2∇u), V is a continuous and coercive potential function with positive infimum, f is a Carathéodory function defined on R3 × R satisfying the classic Ambrosetti-Rabinowitz condition. First, a nontrivial solution is obtained for λ small enough and ε fixed by variational methods and truncation technique. Later, the asymptotic behaviour of these solutions is studied whenever ε and λ tend to zero respectively. We prove that they converge to a nontrivial solution of a classic Schrödinger-Poisson system and a class of Schrödinger equation associated respectively