Normalized solution to the nonlinear p-Laplacian equation with an L^2 constrain: mass supercritical case

TL;DR

This paper investigates the existence of ground state solutions for a p-Laplacian equation with an L^2 constraint in the mass supercritical case, extending known results for the scalar field equation and relaxing nonlinearity conditions.

Contribution

It introduces a normalized solution framework for the p-Laplacian with L^2 constraints, broadening the understanding of solutions in the mass supercritical regime and generalizing previous scalar field results.

Findings

Existence of ground state solutions under mass supercritical conditions

Analysis of ground state energy behavior with varying mass

Extension of scalar field equation results to p-Laplacian case

Abstract

In this paper, we study the existence of ground state solutions to the following p-Laplacian equation in some dimension $N\geq3$ with an $L^2$ constraint: \begin{equation*} \begin{cases} -\Delta_{p}u+{\vert u\vert}^{p-2}u=f(u)-\mu u \quad \text{ in } \mathbb{R}^N,\\ {\Vert u\Vert}^2_{L^2(\mathbb{R}^N)}=m,\\ u\in W^{1,p}(\mathbb{R}^N)\cap L^2(\mathbb{R}^N), \end{cases} \end{equation*} where $-\Delta_{p}u=div\left( {\vert\nabla u\vert}^{p-2}\nabla u \right)$, $2\leq p<N$, $f\in C(\mathbb{R},\mathbb{R})$, $m>0$, $\mu\in\mathbb{R}$ will appear as a Lagrange multiplier and the continuous nonlinearity $f$ satisfies mass supercritical conditions. We mainly study the behavior of ground state energy $E_m$ with $m>0$ changing within a certain range and aim at extending nonlinear scalar field equation when $p=2$ and reducing the constraint condition of nonlinearity $f$.