Multiplicity and concentration of solutions to a fractional p-Laplace problem with exponential growth

Abstract

In this paper, we study the Schrödinger equation involving $\frac{N}{s}$-fractional Laplace as follows ${\epsilon }^N(-\mathrm{\Delta }{)}_{N/s}^{s}u+V(x)|u{|}^{\frac{N}{s}-2}u=f(u)$ in ${\mathbb{R}}^N$, where $\epsilon$ is a positive parameter, $N=ps$, $s\in (0,1)$. The nonlinear function $f$ has the exponential growth and potential function $V$ is a continuous function satisfying some suitable conditions. Our problem lacks of compactness. By using the Ljusternik-Schnirelmann theory, we obtain the existence, multiplicity and concentration of nontrivial nonnegative solutions for small values of the parameter.