The existence of sign-changing solution for a class of quasilinear Schr\xf6dinger\u2013Poisson systems via perturbation method

Lizhen Chen,Xiaojing Feng,Xinan Hao

doi:10.1186/s13661-019-1272-3

Abstract

This paper is concerned with the existence of a sign-changing solution to a class of quasilinear Schrödinger–Poisson systems. There are some technical difficulties in applying variational methods directly to the problem because the quasilinear term makes it impossible to find a suitable space in which the corresponding functional possesses both smoothness and compactness properties. In order to overcome the difficulties caused by nonlocal term and quasi-linear term, we shall apply the perturbation method by adding a 4-Laplacian operator to consider the perturbation problem. And then, by using the approximation technique, a sign-changing solution with precisely two nodal domains is derived.

Highlights

Introduction and main resultsIn this paper, we consider the existence of a sign-changing solution for the following system: ⎧ ⎨– u + V (x)u φu1 2 u u2 = f (u), in R3,⎩– φ = u2, in R3, (1.1)where V is a continuous potential function and f is an appropriate nonlinear function
We consider the existence of a sign-changing solution for the following system:
According to a classical model, the interaction of a charge particle with an electromagnetic field can be described by coupling the nonlinear Schrödinger equations and Poisson equations

Summary

Introduction and main results

We consider the existence of a sign-changing solution for the following system:. Wang and Zhou in [39] obtained a sign-changing solution for system (1.2) by seeking minimizer of the energy functional I over the following constraint: M0 = u ∈ H1 R3 : u± = 0, I (u), u+ = I (u), u– = 0. This argument mainly shows that there is a minimizer of I constrained on M0 and verifies that the minimizer is a critical point of I via quantitative deformation lemma and degree theory.

Preliminary

HV1 v–λ