{Localized nodal solutions for $p-$Laplacian equations with critical exponents in ${\mathbb{R}}^N$

Abstract

In this paper, we consider the existence of localized sign-changing solutions for the $p-$Laplacian nonlinear Schrödinger equation $$-{\epsilon }^p{\Delta }_{p}u+V(x)|u{|}^{p-2}u=|u{|}^{p^{\ast }-2}u+\mu |u{|}^{q-2}u,u\in W^{1,p}({\mathbb{R}}^N),$$ where $1<p<N$, $p_N=\max \{p,p^{\ast }-1\}<q<p^{\ast }=\frac{Np}{N-p}$, $\mu >0$, ${\Delta }_p$ is the $p-$Laplacian operator. By using the penalization method together with the truncation method and a blow-up argument, we establish for small $\epsilon$ the existence of a sequence of localized nodal solutions concentrating near a given local minimum point of the potential function.