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

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

paper · doi:10.48550/arxiv.2508.16010

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), · amp; 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

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