Ground state and multiple solutions for modified autonomous fourth-order elliptic equations with Berestycki-Lions type conditions

TL;DR

This paper proves the existence of a ground state and infinitely many solutions for a modified fourth-order elliptic equation with Berestycki-Lions type nonlinearity, using novel variational methods.

Contribution

It introduces a new approach combining Jeanjean's technique with Pohozaev-Palais-Smale sequences for ground states and establishes infinite solutions for odd nonlinearities.

Findings

Existence of a ground state solution.

Infinitely many radially symmetric solutions when f is odd.

Resolution of previous gaps under weak nonlinearity conditions.

Abstract

This article establishes the existence of a ground state and infinitely many solutions for the modified fourth-order elliptic equation: \[ \begin{aligned} \left\{ \begin{array}{ll} \Delta^2 u - \Delta u + u - \frac{1}{2}u\Delta(u^2) = f(u), & \text{in } \mathbb{R}^N, u \in H^2(\mathbb{R}^N), \end{array} \right. \end{aligned} \] where $4 < N \leq 6$ and$f:\mathbb{R}\rightarrow\mathbb{R}$ is a nonlinearity of Berestycki-Lions type. For the ground state solution, we develop a novel approach that combines Jeanjean's technique with a Pohozaev-Palais-Smale sequence construction. When $f$ is odd, we prove infinite multiplicity of radially symmetric solutions via minimax methods on a topologically constrained comparison functional. This work resolves the lack of results for this autonomous problem under almost the weakest nonlinearity conditions.