On the Variable-order Fractional Laplacian Equation with Variable Growth on $\mathbb{R}^N$

Abstract

The aim of this paper is to study the existence of solutions to the variable-order fractional Laplacian as follows: \[ (-\Delta)^{s(\cdot)}u + V(x)u = \lambda f(x,u) \quad \textrm{in $\mathbb{R}^N$}, \] where $\lambda \gt 0$ is a parameter, $N \geq 1$, $(-\Delta)^{s(\cdot)}$ is the variable-order fractional Laplacian operator with $s(\cdot) \colon \mathbb{R}^N \times \mathbb{R}^N \to (0,1)$ is continuous function with $N \gt 2s^{+} \geq 2s(x,y)$ for all $(x,y) \in \mathbb{R}^N \times \mathbb{R}^N$, and $f$ has variable growth and $V$ satisfies some suitable assumptions. Using Mountain Pass Theorem, Fountain Theorem and Genus theory, we obtain the existence of weak solutions to above problem.