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

• Chongqing Wei ^[1] ; Anran Li ^[1] ; Leiga Zhao ^[2]
  1. [1] Shanxi University

     Shanxi University

     China

     
  2. [2] Beijing Technology and Business University

     Beijing Technology and Business University

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 21, Nº 3, 2022
• Idioma: inglés
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• Resumen

  • 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