Multiplicity and concentration results for a (p, q)-Laplacian problem in RN

In this paper, we study the multiplicity and concentration of positive solutions for the following ( p , q )-Laplacian problem: - Δ p u - Δ q u + V ( ε x ) | u | p - 2 u + | u | q - 2 u = f ( u ) in R N , u ∈ W 1 , p ( R N ) ∩ W 1 , q ( R N ) , u > 0 in R N , where ε > 0 is a small parameter, 1 < p < q < N , Δ r u = div ( | ∇ u | r - 2 ∇ u ) , with r ∈ { p , q } , is the r -Laplacian operator, V : R N → R is a continuous function satisfying the global Rabinowitz condition, and f : R → R is a continuous function with subcritical growth. Using suitable variational arguments and Ljusternik–Schnirelmann category theory, we investigate the relation between the number of positive solutions and the topology of the set where V attains its minimum for small ε .