Existence and multiplicity of normalized solutions for $(2,q)$-Laplacian equations with generic double-behaviour nonlinearities
Rui Ding, Chao Ji, Patrizia Pucci
TL;DR
This work addresses the existence and multiplicity of normalized solutions for a (2,q)-Laplacian equation with a nonlinearity f exhibiting mass-subcritical behavior near the origin and mass-supercritical behavior at infinity. The authors formulate a constrained variational problem on the mass sphere and exploit a Pohozaev-type framework to obtain two distinct normalized solutions: a negative-energy, locally minimal solution and a second, higher-energy solution. A key novelty is the splitting of the nonlinearity and energy into components with different growth properties, enabling precise control of the Nehari and Pohozaev constraints and the construction of a second solution via a constrained minimization on a Pohozaev manifold. The results extend the theory of normalized solutions to the mixed (2,q)-Laplacian setting and provide concrete examples and conditions under which the second solution exists, including parity-based symmetry results for the second solution.
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 $1<q<N$, $N\geq3$, $Δ_q=\operatorname{div}\left(|\nabla u|^{q-2} \nabla u\right)$ denotes the $q$-Laplacian operator, $λ$ 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.