| Autor: |
Mónica Clapp, Filomena Pacella |
| Jazyk: |
angličtina |
| Rok vydání: |
2021 |
| Předmět: |
|
| Zdroj: |
Mathematics in Engineering, Vol 3, Iss 3, Pp 1-15 (2021) |
| Druh dokumentu: |
article |
| ISSN: |
2640-3501 |
| DOI: |
10.3934/mine.2021022?viewType=HTML |
| Popis: |
We study the critical Neumann problem \begin{equation*} \begin{cases} -\Delta u = |u|^{2^*-2}u &\text{in }\Sigma_\omega,\\ \quad\frac{\partial u}{\partial\nu}=0 &\text{on }\partial\Sigma_\omega, \end{cases} \end{equation*} in the unbounded cone $\Sigma_\omega:=\{tx:x\in\omega\text{ and }t>0\}$, where $\omega$ is an open connected subset of the unit sphere $\mathbb{S}^{N-1}$ in $\mathbb{R}^N$ with smooth boundary, $N\geq 3$ and $2^*:=\frac{2N}{N-2}$. We assume that some local convexity condition at the boundary of the cone is satisfied. If $\omega$ is symmetric with respect to the north pole of $\mathbb{S}^{N-1}$, we establish the existence of a nonradial sign-changing solution. On the other hand, if the volume of the unitary bounded cone $\Sigma_\omega\cap B_1(0)$ is large enough (but possibly smaller than half the volume of the unit ball $B_1(0)$ in $\mathbb{R}^N$), we establish the existence of a positive nonradial solution. |
| Databáze: |
Directory of Open Access Journals |
| Externí odkaz: |
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