Existence and multiplicity of normalized solutions for $(2,q)$-Laplacian equations with generic double-behaviour nonlinearities

TL;DR

This paper investigates the existence and multiplicity of normalized solutions for a $(2,q)$-Laplacian equation with general nonlinearities that exhibit double-behavior, providing new results under various assumptions.

Contribution

It establishes the existence of both a least-energy solution and a higher-energy solution for the $(2,q)$-Laplacian equation with more general nonlinearities than previously considered.

Findings

Existence of a locally least-energy solution.

Existence of a second, higher-energy solution.

Results hold under various assumptions on the nonlinearity.

Abstract

In this paper, we study {existence and multiplicity} of normalized solutions for the following $(2, q)$-Laplacian equation \begin{equation*}\label{Eq-Equation1} \left\{\begin{array}{l} -\Delta u-\Delta_q u+\lambda u=f(u) \quad x \in \mathbb{R}^N , \int_{\mathbb{R}^N}u^2 d x=c^2, \end{array}\right. \end{equation*} where $1<q<N$, $N\geq3$, $\Delta_q=\operatorname{div}\left(|\nabla u|^{q-2} \nabla u\right)$ denotes the $q$-Laplacian operator, $\lambda$ is a Lagrange multiplier and $c>0$ is a constant. The nonlinearity $f:\mathbb{R}\rightarrow \mathbb{R}$ is continuous, with mass-subcritical growth at the origin, mass-supercritical growth at infinity, and is more general than the sum of two powers. Under different assumptions, we prove the existence of a locally least-energy solution and the existence of a second solution with higher energy.