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Existence of nonradial positive and nodal solutions to a critical Neumann problem in a cone

  • Received: 22 June 2019 Accepted: 12 October 2019 Published: 15 July 2020
  • 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.

    Citation: Mónica Clapp, Filomena Pacella. Existence of nonradial positive and nodal solutions to a critical Neumann problem in a cone[J]. Mathematics in Engineering, 2021, 3(3): 1-15. doi: 10.3934/mine.2021022

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  • 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.


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