Multiplicity of concentrating solutions for (p, q)-Schrödinger equations with lack of compactness

We study the multiplicity of concentrating solutions for the following class of (p, q)-Laplacian problems (Formula presented.) where ε > 0 is a small parameter, γ∈{0,1},1 infΛV for some bounded open set Λ ⊂ ℝN, and f: ℝ → ℝ is a continuous nonlinearity with subcritical growth. The main results are obtained by combining minimax theorems, penalization technique and Ljusternik–Schnirelmann category theory. We also provide a multiplicity result for a supercritical version of the above problem by combining a truncation argument with a Moser-type iteration. As far as we know, all these results are new.