Ground states to a quasilinear Schr\"{o}dinger equation with Berestycki-Lions type nonlinearities

TL;DR

This paper establishes the existence and multiplicity of ground state and nonradial solutions for a quasilinear Schrödinger equation with Berestycki-Lions type nonlinearities, extending classical results to a quasilinear setting.

Contribution

It introduces a novel critical point approach on a topological manifold to prove solutions for the quasilinear Schrödinger equation with Berestycki-Lions nonlinearities.

Findings

Existence of a ground state for N ≥ 3

Existence of a nonradial ground state for N ≥ 4

Infinitely many nonradial solutions for N = 4 or N ≥ 6

Abstract

In this paper, we study the following quasilinear {S}chr\"{o}dinger equation: $$\left\{ \begin{array}{l} - {\Delta u} - \frac{\kappa }{2}\Delta \left( {u}^{2}\right) u = h\left( u\right) \text{ in }{\mathbb{R}}^{N}, \\ u \in {H}^{1}\left( {\mathbb{R}}^{N}\right), \end{array}\right. $$ where $N \geq 3,\kappa > 0$ is a parameter, and $h$ satisfies Berestycki-Lions condition. By using a critical point theory on a topological manifold, we obtain the existence of a ground state for $N \geq 3$, a nonradial ground state for $N \geq 4$, and infinitely many nonradial solutions for $N = 4$ or $N \geq 6$. Our results generalize several classical works into quasilinear equations.