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

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{${\mathcal{E}}_{\hat{I}}»$} \end{equation} where $1<q<N$, $\hat{I}{”}_q=\mathrm{div}\left(|\nabla u{|}^{q-2}\nabla u\right)$ is the $q$-Laplacian operator, $\hat{I}»$ is a Lagrange multiplier and $c>0$ is a constant. The nonlinearity $g:\mathbb{R}\to \mathbb{R}$ is continuous and the behaviour of $g$ at the origin is allowed to be strongly sublinear, i.e., $\underset{s\to 0}{\lim }g(s)/s=-\infty$, 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.

arXiv admin note: text overlap with arXiv:2306.06015 by other authors