Research article

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

  • 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.

    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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  • 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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