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

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

$$(-{\mathrm{\Delta }}_{p(x)}{)}^{s}u+V(x)|u{|}^{p(x)-2}u=f(x,u)\text{in}{\mathbb{R}}^N,$$

where $(-{\mathrm{\Delta }}_{p(x)}{)}^s$ is the fractional $p(x)$-Laplace operator, $s\in (0,1)$, $V:{\mathbb{R}}^N\to (0,+\mathrm{\infty })$ 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.