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

Abstract

We study the multiplicity of concentrating solutions for the following class of (p, q)-Laplacian problems

$$\left\{{\matrix{{- {\Delta _p}u - {\Delta _q}u + V(\varepsilon \,x)({u^{p - 1}} + {u^{q - 1}}) = f(u) + \gamma {u^{{q^ *} - 1}}\,{\rm{in}}\,{\mathbb{R}^N},} \hfill \cr {u \in {W^{1,p}}({\mathbb{R}^N}) \cap {W^{1,q}}({\mathbb{R}^N}),\,\,u > 0\,\,{\rm{in}}\,\,{\mathbb{R}^N},} \hfill \cr}} \right.$$

where ε > 0 is a small parameter, \(\gamma \in \{0,1\},\,1 < p < q < N,\,\,{q^*} = {{Nq} \over {N - q}}\) is the critical Sobolev exponent, \({\Delta _s}u = {\rm{div}}(|\nabla u{|^{s - 2}}\nabla u)\), with s ∈ {p, q}, is the s-Laplacian operator, V: ℝ^N → ℝ is a positive continuous potential such that inf_∂Λ V > inf_Λ V for some bounded open set Λ ⊂ ℝ^N, 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.