Ground state and multiple solutions for modified autonomous fourth-order elliptic equations with Berestycki-Lions type conditions
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918128899850240 |
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| author | Yin, Lifeng Wang, Fan |
| author_facet | Yin, Lifeng Wang, Fan |
| contents | This article establishes the existence of a ground state and infinitely many solutions for the modified fourth-order elliptic equation:
\[
\begin{aligned}
\left\{
\begin{array}{ll}
Δ^2 u - Δu + u - \frac{1}{2}uΔ(u^2) = f(u), & \text{in } \mathbb{R}^N,
u \in H^2(\mathbb{R}^N),
\end{array}
\right.
\end{aligned}
\]
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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_16010 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Ground state and multiple solutions for modified autonomous fourth-order elliptic equations with Berestycki-Lions type conditions Yin, Lifeng Wang, Fan Analysis of PDEs This article establishes the existence of a ground state and infinitely many solutions for the modified fourth-order elliptic equation: \[ \begin{aligned} \left\{ \begin{array}{ll} Δ^2 u - Δu + u - \frac{1}{2}uΔ(u^2) = f(u), & \text{in } \mathbb{R}^N, u \in H^2(\mathbb{R}^N), \end{array} \right. \end{aligned} \] 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. |
| title | Ground state and multiple solutions for modified autonomous fourth-order elliptic equations with Berestycki-Lions type conditions |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2508.16010 |