Existence and multiplicity of solutions for a Schrödinger type equations involving the fractional $ p(x) $-Laplacian

Shuhai Zhu; Shuhai Zhu

doi:10.3934/math.2023836

AIMS Mathematics

2023, Volume 8, Issue 7: 16320-16339. doi: 10.3934/math.2023836

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Research article

Existence and multiplicity of solutions for a Schrödinger type equations involving the fractional $ p(x) $-Laplacian

Shuhai Zhu ^,

College of Basic Science, Ningbo University of Finance & Economics, Ningbo 315175, China

Received: 24 February 2023 Revised: 25 April 2023 Accepted: 26 April 2023 Published: 08 May 2023
MSC : 35B08, 35J60, 35A15

We are concerned with the following Schrödinger type equation with variable exponents

$ \begin{equation*} (-\Delta_{p(x)})^{s}u+V(x)|u|^{p(x)-2}u = f(x, u)\, \, \, \, \text{in}\, \, \, \, \mathbb{R}^{N}, \end{equation*} $

where $ (-\Delta_{p(x)})^{s} $ is the fractional $ p(x) $-Laplace operator, $ s\in (0, 1) $, $ V:\mathbb{R}^{N}\to (0, +\infty) $ is a continuous potential function, and $ f:\mathbb{R}^{N}\times\mathbb{R}\to \mathbb{R} $ satisfies the Carathéodory condition. We study the nonlinearity of this equation which is superlinear but does not satisfy the Ambrosetti-Rabinowitz type condition. By using variational techniques and the fountain theorem, we obtain the existence and multiplicity of nontrivial solutions. Furthermore, we show that the problem has a sequence of solutions with high energies.
- fractional $ p(x) $-Laplacian,
- fractional Sobolev space with variable exponent,
- variational method,
- fountain theorem
Citation: Shuhai Zhu. Existence and multiplicity of solutions for a Schrödinger type equations involving the fractional $ p(x) $-Laplacian[J]. AIMS Mathematics, 2023, 8(7): 16320-16339. doi: 10.3934/math.2023836

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Abstract

We are concerned with the following Schrödinger type equation with variable exponents

$ \begin{equation*} (-\Delta_{p(x)})^{s}u+V(x)|u|^{p(x)-2}u = f(x, u)\, \, \, \, \text{in}\, \, \, \, \mathbb{R}^{N}, \end{equation*} $

where $ (-\Delta_{p(x)})^{s} $ is the fractional $ p(x) $-Laplace operator, $ s\in (0, 1) $, $ V:\mathbb{R}^{N}\to (0, +\infty) $ is a continuous potential function, and $ f:\mathbb{R}^{N}\times\mathbb{R}\to \mathbb{R} $ satisfies the Carathéodory condition. We study the nonlinearity of this equation which is superlinear but does not satisfy the Ambrosetti-Rabinowitz type condition. By using variational techniques and the fountain theorem, we obtain the existence and multiplicity of nontrivial solutions. Furthermore, we show that the problem has a sequence of solutions with high energies.

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