On existence results for a class of biharmonic elliptic problems without (AR) condition

Dengfeng Lu; Shuwei Dai; Dengfeng Lu; Shuwei Dai

doi:10.3934/math.2024919

AIMS Mathematics

2024, Volume 9, Issue 7: 18897-18909. doi: 10.3934/math.2024919

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Research article

On existence results for a class of biharmonic elliptic problems without (AR) condition

Dengfeng Lu ¹,
Shuwei Dai ^{2
,
,}

1.
School of Mathematics and Statistics, Hubei Engineering University, Xiaogan 432000, China
2.
College of Technology, Hubei Engineering University, Xiaogan 432000, China

Received: 28 January 2024 Revised: 03 March 2024 Accepted: 07 March 2024 Published: 05 June 2024
MSC : 35J35, 35B38, 35J91

In this paper, we study the following biharmonic elliptic equation in $ \mathbb{R}^{N} $:

$ \Delta^{2}\psi-\Delta \psi+P(x)\psi = g(x, \psi), \ \ x\in\mathbb{R}^{N}, $

where $ g $ and $ P $ are periodic in $ x_{1}, \cdots, x_{N} $, $ g(x, \psi) $ is subcritical and odd in $ \psi $. Without assuming the Ambrosetti-Rabinowitz condition, we prove the existence of infinitely many geometrically distinct solutions for this equation, and the existence of ground state solutions is established as well.
- biharmonic elliptic equation,
- (AR) condition,
- critical point theory,
- Nehari manifold,
- infinitely many solutions,
- geometrically distinct solutions
Citation: Dengfeng Lu, Shuwei Dai. On existence results for a class of biharmonic elliptic problems without (AR) condition[J]. AIMS Mathematics, 2024, 9(7): 18897-18909. doi: 10.3934/math.2024919

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Abstract

In this paper, we study the following biharmonic elliptic equation in $ \mathbb{R}^{N} $:

$ \Delta^{2}\psi-\Delta \psi+P(x)\psi = g(x, \psi), \ \ x\in\mathbb{R}^{N}, $

where $ g $ and $ P $ are periodic in $ x_{1}, \cdots, x_{N} $, $ g(x, \psi) $ is subcritical and odd in $ \psi $. Without assuming the Ambrosetti-Rabinowitz condition, we prove the existence of infinitely many geometrically distinct solutions for this equation, and the existence of ground state solutions is established as well.

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