Existence and concentration of positive solutions for a critical p&q equation

Abstract

Abstract We show existence and concentration results for a class of p&q critical problems given by − d i v a ϵ p | ∇ u | p ϵ p | ∇ u | p − 2 ∇ u + V ( z ) b | u | p | u | p − 2 u = f ( u ) + | u | q ⋆ − 2 u in R N , $$-div\left(a\left(\epsilon^{p}|\nabla u|^{p}\right) \epsilon^{p}|\nabla u|^{p-2} \nabla u\right)+V(z) b\left(|u|^{p}\right)|u|^{p-2} u=f(u)+|u|^{q^{\star}-2} u\, \text{in} \,\mathbb{R}^{N},$$ where u ∈ W 1,p (ℝ N ) ∩ W 1,q (ℝ N ), ϵ > 0 is a small parameter, 1 < p ≤ q < N, N ≥ 2 and q * = Nq/(N − q). The potential V is positive and f is a superlinear function of C 1 class. We use Mountain Pass Theorem and the penalization arguments introduced by Del Pino & Felmer’s associated to Lions’ Concentration and Compactness Principle in order to overcome the lack of compactness.