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: {-Δpu-Δqu+V(εx)(|u|p-2u+|u|q-2u)=f(u)inRN,u∈W1,p(RN)∩W1,q(RN),u>0inRN,where ε> 0 is a small parameter, 1 < p< q< N, Δru=div(|∇u|r-2∇u), with r∈ { p, q} , is the r-Laplacian operator, V: RN→ 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.