Localized nodal solutions for p-Laplacian equations with critical exponents

Abstract

In this paper, we consider the following p-Laplacian Schrödinger equation with a critical exponent −εpΔpv+V(x)|v|p−2v=|v|p*−2v+μ|v|q−2v, v∈W1,p(RN), where 1 < p < N, pN=max{p,p*−1}<q<p*=NpN−p, μ > 0 is a constant, ɛ > 0 is a small parameter, and Δpv ≔ ∇ · (|∇v|p−2∇v) is the p-Laplacian operator. By using the penalization method together with the truncation technique and blow-up arguments, we establish, for ɛ small, the existence of a sequence of localized nodal solutions concentrating near a given local minimum point of the potential function V(x).