Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2407.11415 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866916326074744832 |
|---|---|
| author | Zhang, Weiqiang Wen, Yanyun |
| author_facet | Zhang, Weiqiang Wen, Yanyun |
| contents | We are devoted to the study of the following nonlinear $p$-Laplacian Schrödinger equation with $L^{p}$-norm constraint \begin{align*} \begin{cases} &-Δ_{p} u=λ|u|^{p-2}u +|u|^{r-2}u\quad\mbox{in}\quadΩ,\\ &u=0\quad\mbox{on}\quad \partialΩ,\\ &\int_Ω|u|^{p}dx=a, \end{cases} \end{align*} where $Δ_{p}u=\text{div} (|\nabla u|^{p-2}\nabla u)$, $Ω\subset\mathbb{R}^{N}$ is an exterior domain with smooth boundary $\partialΩ\neq\emptyset$ satisfying that $\R^{N}\setminusΩ$ is bounded, $N\geq3$, $2\leq p<N$, $p<r<p+\frac{p^{2}}{N}$, $a>0$ and $λ\in\R$ is an unknown Lagrange multiplier. First, by using the splitting techniques and the Gagliardo-Nirenberg inequality, the compactness of Palais-Smale sequence of the above problem at higher energy level is established. Then, exploiting barycentric function methods, Brouwer degree and minimax principle, we obtain a solution $(u,\la)$ with $u>0$ in $\R^{N}$ and $\la<0$ when $\R^{N}\setminusΩ$ is contained in a small ball. Moreover, we give a similar result if we remove the restriction on $Ω$ and assume $a>0$ small enough. Last, with the symmetric assumption on $Ω$, we use genus theory to consider infinite many solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_11415 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Normalized solution for $p$-Laplacian equation in exterior domain Zhang, Weiqiang Wen, Yanyun Analysis of PDEs We are devoted to the study of the following nonlinear $p$-Laplacian Schrödinger equation with $L^{p}$-norm constraint \begin{align*} \begin{cases} &-Δ_{p} u=λ|u|^{p-2}u +|u|^{r-2}u\quad\mbox{in}\quadΩ,\\ &u=0\quad\mbox{on}\quad \partialΩ,\\ &\int_Ω|u|^{p}dx=a, \end{cases} \end{align*} where $Δ_{p}u=\text{div} (|\nabla u|^{p-2}\nabla u)$, $Ω\subset\mathbb{R}^{N}$ is an exterior domain with smooth boundary $\partialΩ\neq\emptyset$ satisfying that $\R^{N}\setminusΩ$ is bounded, $N\geq3$, $2\leq p<N$, $p<r<p+\frac{p^{2}}{N}$, $a>0$ and $λ\in\R$ is an unknown Lagrange multiplier. First, by using the splitting techniques and the Gagliardo-Nirenberg inequality, the compactness of Palais-Smale sequence of the above problem at higher energy level is established. Then, exploiting barycentric function methods, Brouwer degree and minimax principle, we obtain a solution $(u,\la)$ with $u>0$ in $\R^{N}$ and $\la<0$ when $\R^{N}\setminusΩ$ is contained in a small ball. Moreover, we give a similar result if we remove the restriction on $Ω$ and assume $a>0$ small enough. Last, with the symmetric assumption on $Ω$, we use genus theory to consider infinite many solutions. |
| title | Normalized solution for $p$-Laplacian equation in exterior domain |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2407.11415 |