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

TL;DR

This paper establishes the existence and multiplicity of normalized solutions for a class of $(2, q)$-Laplacian equations with strongly sublinear nonlinearities, including logarithmic cases, using approximation methods and variational techniques.

Contribution

It introduces a novel approach to find normalized solutions for $(2, q)$-Laplacian equations with strongly sublinear nonlinearities, including the logarithmic case, and proves the existence of infinitely many solutions.

Findings

Existence of least-energy solutions via approximation methods.

Convergence of approximate solutions to the original problem.

Existence of infinitely many solutions under certain conditions.

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} &-\Delta u-\Delta_q u+\lambda u=g(u),\quad x \in \mathbb{R}^N, &\int_{\mathbb{R}^N}u^2 d x=c^2, \end{aligned}\right. \tag{$\mathscr E_\lambda$} \end{equation} where $1<q<N$, $\Delta_q=\operatorname{div}\left(|\nabla u|^{q-2} \nabla u\right)$ is the $q$-Laplacian operator, $\lambda$ 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.