LAUSR

Abstract In this paper, we study the Schrodinger equation involving the fractional ( p , p 1 ) -Laplacian and Trudinger-Moser nonlinearity as follows ( − Δ ) N / s s u + ( − Δ ) p 1 s u + V ( e x ) ( | u | N s − 2 u + | u | p 1 − 2 u ) = f ( u ) in R N , where e is a positive parameter, N = p s , s ∈ ( 0 , 1 ) , 2 ≤ N s = p p 1 . The nonlinear function f has the exponential growth and potential function V is continuous function satisfying some suitable conditions. By using the Ljusternik-Schnirelmann theory, we obtain the existence, multiplicity and concentration of nontrivial nonnegative solutions for small values of the parameter. In our best knowledge, it is the first time that the fractional ( p , p 1 ) -Laplacian problem involving exponential growth is studied.