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 \(\varepsilon^{N}(-\Delta)_{N/s}^{s}u+V(x)|u|^{\frac{N}{s}-2}u=f(u)\) in \(\mathbb R^{N}\), where \(\varepsilon\) 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.