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

TL;DR

This work analyzes $L^2$-normalized solutions to a mixed diffusion equation $-\Delta u - \Delta_q u + \lambda u = \alpha |u|^{p-2}u$ in $\mathbb{R}^N$ with $p$ between the mass-critical exponents $p_2$ and $p_q$, focusing on the intermediate regime. Using a variational framework on the mass sphere, the authors establish negative-energy global minimizers, positive-energy local minimizers, and a mountain-pass solution, all under appropriate parameter ranges, along with a comprehensive treatment of the zero-mass case $\lambda=0$ via Liouville-type arguments. They prove existence, nonexistence, and multiplicity results, including radial symmetry and regularity, and derive decay estimates for zero-mass radial solutions, extending the understanding of $(2,q)$-Laplacian models. The results rely on compactness up to translations, Pohozaev identities, energy estimates, and Ekeland’s principle, contributing to the theory of normalized solutions in mixed diffusion settings. These findings illuminate how mixed diffusion shapes the qualitative behavior of normalized states in intermediate nonlinear regimes.

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 $λ= 0$, 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.

Theorems & Definitions (27)

• Theorem 1.1
• Remark 1
• Theorem 1.2
• Remark 2
• Theorem 1.3
• Proposition 1
• Theorem 1.4
• Lemma 1
• Lemma 2
• Theorem 3.1