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

TL;DR

This paper proves the existence of localized, sign-changing solutions for a $p$-Laplacian nonlinear Schrödinger equation with critical exponents, using penalization, truncation, and blow-up techniques for small $psilon$.

Contribution

It introduces a novel combination of penalization, truncation, and blow-up methods to establish localized nodal solutions in the critical exponent setting.

Findings

Existence of localized nodal solutions near potential minima for small epsilon.

Solutions concentrate around specific points as epsilon approaches zero.

Method extends to equations with critical exponents and sign-changing solutions.

Abstract

In this paper, we consider the existence of localized sign-changing solutions for the $p-$Laplacian nonlinear Schr\"odinger equation $$ -\epsilon^p\Delta_pu+V(x)|u|^{p-2}u=|u|^{p^*-2}u+\mu|u|^{q-2}u,~~u\in W^{1,p}(\mathbb{R}^N), $$ where $1<p<N$, $p_N=\max\{p,p^*-1\}<q<p^*=\frac{Np}{N-p}$, $\mu>0$, $\Delta_p$ 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.