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

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} -Δu-Δ_q u+λu=f(u) \quad x \in \mathbb{R}^N , \int_{\mathbb{R}^N}u^2 d x=c^2, \end{array}\right. \end{equation*} where $1<q<N$, $N\ge 3$, $\hat{I}{”}_q=\mathrm{div}\left(|\nabla u{|}^{q-2}\nabla u\right)$ denotes the $q$-Laplacian operator, $\hat{I}»$ is a Lagrange multiplier and $c>0$ is a constant. The nonlinearity $f:\mathbb{R}\to \mathbb{R}$ 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.

arXiv admin note: substantial text overlap with arXiv:2405.05194 by other authors