Infinitely many solutions for a class of biharmonic equations with indefinite potentials

Wen Guan; Da-Bin Wang; Xinan Hao; Wen Guan; Da-Bin Wang; Xinan Hao

doi:10.3934/math.2020235

AIMS Mathematics

2020, Volume 5, Issue 4: 3634-3645. doi: 10.3934/math.2020235

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Research article

Infinitely many solutions for a class of biharmonic equations with indefinite potentials

1.
Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu, 730050, P. R. China
2.
School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong, 273165, P. R. China

Received: 12 February 2020 Accepted: 10 April 2020 Published: 16 April 2020
MSC : 35J20, 35J65

In this paper, we consider the following sublinear biharmonic equations $ \begin{equation*} \Delta^2 u + V\left( x \right)u = K(x)|u|^{p-1}u,\ x\in \mathbb{R}^N, \end{equation*} $ where $N\geq5, ~0 \lt p \lt 1$, and $K, V$ both change sign in $\mathbb{R}^N$. We prove that the problem has infinitely many solutions under appropriate assumptions on $K, V$. To our end, we firstly infer the boundedness of $PS$ sequence, and then prove that the $PS$ condition was satisfied. At last, we verify that the corresponding functional satisfies the conditions of the symmetric Mountain Pass Theorem.
- biharmonic equation,
- indefinite potential,
- symmetric Mountain Pass Theorem
Citation: Wen Guan, Da-Bin Wang, Xinan Hao. Infinitely many solutions for a class of biharmonic equations with indefinite potentials[J]. AIMS Mathematics, 2020, 5(4): 3634-3645. doi: 10.3934/math.2020235

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Abstract

In this paper, we consider the following sublinear biharmonic equations $ \begin{equation*} \Delta^2 u + V\left( x \right)u = K(x)|u|^{p-1}u,\ x\in \mathbb{R}^N, \end{equation*} $ where $N\geq5, ~0 \lt p \lt 1$, and $K, V$ both change sign in $\mathbb{R}^N$. We prove that the problem has infinitely many solutions under appropriate assumptions on $K, V$. To our end, we firstly infer the boundedness of $PS$ sequence, and then prove that the $PS$ condition was satisfied. At last, we verify that the corresponding functional satisfies the conditions of the symmetric Mountain Pass Theorem.

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