Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2406.07985 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910833494196224 |
|---|---|
| author | Ding, Rui Ji, Chao Pucci, Patrizia |
| author_facet | Ding, Rui Ji, Chao Pucci, Patrizia |
| contents | 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{$\mathscr E_λ$} \end{equation} where $1<q<N$, $Δ_q=\operatorname{div}\left(|\nabla u|^{q-2} \nabla u\right)$ is the $q$-Laplacian operator, $λ$ is a Lagrange multiplier and $c>0$ is a constant. The nonlinearity $g:\mathbb{R}\rightarrow \mathbb{R}$ is continuous and the behaviour of $g$ at the origin is allowed to be strongly sublinear, i.e., $\lim \limits _{s \rightarrow 0} g(s) / s=-\infty$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_07985 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Normalized solutions to a class of $(2, q)$-Laplacian equationsin the strongly sublinear regime Ding, Rui Ji, Chao Pucci, Patrizia Analysis of PDEs 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{$\mathscr E_λ$} \end{equation} where $1<q<N$, $Δ_q=\operatorname{div}\left(|\nabla u|^{q-2} \nabla u\right)$ is the $q$-Laplacian operator, $λ$ is a Lagrange multiplier and $c>0$ is a constant. The nonlinearity $g:\mathbb{R}\rightarrow \mathbb{R}$ is continuous and the behaviour of $g$ at the origin is allowed to be strongly sublinear, i.e., $\lim \limits _{s \rightarrow 0} g(s) / s=-\infty$, 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. |
| title | Normalized solutions to a class of $(2, q)$-Laplacian equationsin the strongly sublinear regime |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2406.07985 |