Normalized solutions to a class of $(2, q)$-Laplacian equationsin the strongly sublinear regime

TL;DR

The paper investigates normalized solutions to the $(2,q)$-Laplacian equation with a mass constraint, allowing strongly sublinear nonlinearities including logarithmic terms. It introduces a perturbed energy functional to overcome ill-posedness on the natural function space, proves the existence of least-energy normalized solutions for large mass, and shows that these solutions are sign-definite, radial (up to translation), and radially monotone. Furthermore, under structural decompositions of the nonlinearity and symmetry restrictions, the authors establish the existence of infinitely many normalized solutions via a genus-based minimax argument on symmetric subspaces. The results extend the theory of normalized solutions for mixed $(p,q)$-Laplacian operators to strongly sublinear regimes and provide insight into energy behavior and multiplicity under mass constraints.

Abstract

In this paper, we consider the existence and multiplicity of normalized solutions for the following $(2, q)$-Laplacian equation \begin{equation}\label{Equation1} \left\{\begin{aligned} &-Δu-Δ_q u+λu=g(u),\quad x \in \mathbb{R}^N, &\int_{\mathbb{R}^N}u^2 d x=c^2, \end{aligned}\right. \tag{$\mathscr E_λ$} \end{equation} where $1<q<N$, $Δ_q=\operatorname{div}\left(|\nabla u|^{q-2} \nabla u\right)$ is the $q$-Laplacian operator, $λ$ is a Lagrange multiplier and $c>0$ is a constant. The nonlinearity $g:\mathbb{R}\rightarrow \mathbb{R}$ is continuous and the behaviour of $g$ at the origin is allowed to be strongly sublinear, i.e., $\lim \limits _{s \rightarrow 0} g(s) / s=-\infty$, which includes the logarithmic nonlinearity $$ g(s)= s \log s^2. $$ We consider a family of approximating problems that can be set in $H^1\left(\mathbb{R}^N\right)\cap D^{1, q}\left(\mathbb{R}^N\right)$ and the corresponding least-energy solutions. Then, we prove that such a family of solutions converges to a least-energy solution to the original problem. Additionally, under certain assumptions about $g$ that allow us to work in a suitable subspace of $H^1\left(\mathbb{R}^N\right)\cap D^{1, q}\left(\mathbb{R}^N\right)$, we prove the existence of infinitely many solutions of the above $(2, q)$-Laplacian equation.

Theorems & Definitions (56)

• Theorem 1.1
• Remark 1.2
• Proposition 1.3
• Proposition 1.4
• Theorem 1.5
• Remark 1.6
• Theorem 1.7
• Lemma 2.1: The Gagliardo-Nirenberg inequality, Weinstein1983GN
• Lemma 2.2: $L^q$-Gagliardo-Nirenberg inequality, agueH3008sharp
• Lemma 2.3: The Trudinger-Moser inequality, Cao1992TM