Normalized Solutions for the $(2,q)$-Laplacian Operator Between Mass-Critical Exponents

TL;DR

This paper investigates the existence and nonexistence of normalized solutions for a class of $(2,q)$-Laplacian equations with power nonlinearities, focusing on the intermediate regime between mass critical exponents and highlighting the role of mixed diffusion.

Contribution

It provides a comprehensive analysis of solutions for the $(2,q)$-Laplacian operator in intermediate regimes, combining variational methods and energy estimates, which is novel in this context.

Findings

Existence of solutions with negative energy via global minimization.

Existence of solutions with positive energy via local minimization and mountain-pass.

Nonexistence results for the zero-mass case $oxed{ ext{highlighting the role of mixed diffusion}}$.

Abstract

This paper concerns the existence of normalized solutions to a class of $(2,q)$-Laplacian equations with a power type nonlinearity in the intermediate regime between the two mass critical exponents $2(1+2/N)$, $q(1+2/N)$. More precisely, we prove the existence of solutions with negative energy obtained through a global minimization procedure, and of solutions with positive energy established via a local minimization technique and a mountain-pass argument. Furthermore, we derive both existence and nonexistence results for the zero-mass case $\lambda = 0$, highlighting the role of the mixed diffusion in determining the qualitative behavior of solutions. Specifically, this paper's novelty lies in providing a comprehensive understanding of the intermediate cases that arise when the non-homogeneous $(2,q)$-Laplacian operator appears. Our analysis combines variational methods, compactness arguments, and delicate energy estimates adapted to the nonhomogeneous nature of the $(2,q)$-Laplacian operator.