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.
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