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

Autor: Shuhai Zhu
Jazyk: angličtina
Rok vydání: 2023
Předmět:
Zdroj: AIMS Mathematics, Vol 8, Iss 7, Pp 16320-16339 (2023)
Druh dokumentu: article
ISSN: 2473-6988
DOI: 10.3934/math.2023836?viewType=HTML
Popis: 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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