Research article

Existence of multiple non-trivial solutions for a nonlocal problem

  • Received: 08 January 2019 Accepted: 12 March 2019 Published: 25 March 2019
  • In this paper, we are concerned with the following general nonlocal problem $ \begin{equation*} \begin{cases} -\mathcal{L}_K u = \lambda_1u+f(x, u)& \text{in}\ \Omega, \\ u = 0& \text{in}\ \mathbb{R}^N\backslash\Omega, \end{cases} \end{equation*} $ where $\lambda_1$ denotes the first eigenvalue of the nonlocal integro-differential operator $-\mathcal{L}_K$, $\Omega\subset\mathbb{R}^N$ is open, bounded domain with smooth boundary. Under several structural assumptions on $f$, we verify that the problem possesses at least two non-trivial solutions and locate the region in different parts of the Hilbert space by variational method. As a particular case, we derive an existence theorem for the following equation driven by the fractional Laplacian $ \begin{equation*} \begin{cases} (-\Delta)^su = \lambda_1u+f(x, u)& \text{in}\ \Omega, \\ u = 0& \text{in}\ \mathbb{R}^N\backslash\Omega. \end{cases} \end{equation*} $

    Citation: Xianyong Yang, Zhipeng Yang. Existence of multiple non-trivial solutions for a nonlocal problem[J]. AIMS Mathematics, 2019, 4(2): 299-307. doi: 10.3934/math.2018.2.299

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  • In this paper, we are concerned with the following general nonlocal problem $ \begin{equation*} \begin{cases} -\mathcal{L}_K u = \lambda_1u+f(x, u)& \text{in}\ \Omega, \\ u = 0& \text{in}\ \mathbb{R}^N\backslash\Omega, \end{cases} \end{equation*} $ where $\lambda_1$ denotes the first eigenvalue of the nonlocal integro-differential operator $-\mathcal{L}_K$, $\Omega\subset\mathbb{R}^N$ is open, bounded domain with smooth boundary. Under several structural assumptions on $f$, we verify that the problem possesses at least two non-trivial solutions and locate the region in different parts of the Hilbert space by variational method. As a particular case, we derive an existence theorem for the following equation driven by the fractional Laplacian $ \begin{equation*} \begin{cases} (-\Delta)^su = \lambda_1u+f(x, u)& \text{in}\ \Omega, \\ u = 0& \text{in}\ \mathbb{R}^N\backslash\Omega. \end{cases} \end{equation*} $


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