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

2025/08/22 by Lifeng Yin, Fan Wang, Yin, Lifeng +1
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP

paper · pdf · doi:10.48550/arxiv.2508.16010

arxiv created 2026/08/01 · arxiv updated 2026/08/04

Abstract

This article establishes the existence of a ground state and infinitely many solutions for the modified fourth-order elliptic equation: \beginaligned \ Δ² u - Δu + u - (1)/(2)uΔ(u²) = f(u), · in ℝ^N, u ∈ H²(ℝ^N), . \endaligned where 4 < N ≤ 6 andf:ℝ→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.

Citations

• Concentration on the Boundary and Sign-Changing Solutions for a Slightly Subcritical Biharmonic Problem
• Biharmonic nonlinear scalar field equations
• Nonlinear scalar field equations, I existence of a ground state

Related

• Classical Sobolev approach for a critical fourth-order Leray-Lions type problem: existence and multiplicity of solutions
• A global minimization trick to solve some classes of Berestycki-Lions type problems
• On symmetry and uniqueness of ground states for linear and nonlinear elliptic PDEs
• Multiplicity results for ground state solutions of a semilinear equation via abrupt changes in magnitude of the nonlinearity
• Singularly perturbed Choquard equations with nonlinearity satisfying Berestycki-Lions assumptions