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

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.

Keywords

• fractional $ p(x) $-laplacian
• fractional sobolev space with variable exponent
• variational method
• fountain theorem