Existence and multiplicity of normalized solutions for $(2,q)$-Laplacian equations with generic double-behaviour nonlinearities
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929756674457600 |
|---|---|
| author | Ding, Rui Ji, Chao Pucci, Patrizia |
| author_facet | Ding, Rui Ji, Chao Pucci, Patrizia |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_15066 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Existence and multiplicity of normalized solutions for $(2,q)$-Laplacian equations with generic double-behaviour nonlinearities Ding, Rui Ji, Chao Pucci, Patrizia Analysis of PDEs 35A15, 35B09, 35B38, 35J9 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. |
| title | Existence and multiplicity of normalized solutions for $(2,q)$-Laplacian equations with generic double-behaviour nonlinearities |
| topic | Analysis of PDEs 35A15, 35B09, 35B38, 35J9 |
| url | https://arxiv.org/abs/2410.15066 |