INFINITELY MANY SOLUTIONS FOR A NONLOCAL PROBLEM

Zhi-Yun Tang; Zeng-Qi Ou

doi:10.11948/20190286

2020 Volume 10 Issue 5

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Zhi-Yun Tang, Zeng-Qi Ou. INFINITELY MANY SOLUTIONS FOR A NONLOCAL PROBLEM[J]. Journal of Applied Analysis & Computation, 2020, 10(5): 1912-1917. doi: 10.11948/20190286

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Zhi-Yun Tang, Zeng-Qi Ou. INFINITELY MANY SOLUTIONS FOR A NONLOCAL PROBLEM[J]. Journal of Applied Analysis & Computation, 2020, 10(5): 1912-1917. doi: 10.11948/20190286

INFINITELY MANY SOLUTIONS FOR A NONLOCAL PROBLEM

Zhi-Yun Tang^1, ,,
Zeng-Qi Ou^1, ,

1.
School of Mathematics and Statistics, Southwest University, Tiansheng Road, 400710, Beibei, Chongqing, China

Corresponding authors: Email address: yvonne50@qq.com(Z. Tang); Email address: ouzengq707@sina.com (Z. Ou)

Abstract

Abstract

Consider a class of nonlocal problems $\left\{\begin{array}{lr} -\left(a-b \int_{\Omega}|\nabla u|^{2} d x\right) \Delta u=f(x, u), & x \in \Omega, \\ u=0, & x \in \partial \Omega, \end{array}\right.$ where $a>0, b>0$ , $\Omega\subset \mathbb{R}^N$ is a bounded open domain, $f:\overline{\Omega} \times \mathbb R \longrightarrow \mathbb R$ is a Carath $\acute{\mbox{e}}$ odory function. Under suitable conditions, the equivariant link theorem without the $(P.S.)$ condition due to Willem is applied to prove that the above problem has infinitely many solutions, whose energy increasingly tends to $a^2/(4b)$ , and they are neither large nor small.
- Nonlocal problems /
- infinitely many solutions /
- the $(P.S.)_c$ condition /
- the equivariant link theorem
MSC: 35G20, 35J60, 35J75

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