### Mathematics in Engineering

2021, Issue 3:1-15. doi: 10.3934/mine.2021022
Research article Special Issues

# 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

### Related Papers:

• 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.

 [1] Adimurthi A, Mancini G (1991) The Neumann problem for elliptic equations with critical nonlinearity. Nonlinear Anal, Sc. Norm. Super. di Pisa Quaderni, Scuola Norm. Sup., Pisa, 9-25. [2] Clapp M (2016) Entire nodal solutions to the pure critical exponent problem arising from concentration. J Differ Equations 261: 3042-3060. [3] del Pino M, Musso M, Pacard F, et al. (2011) Large energy entire solutions for the Yamabe equation. J Differ Equations 251: 2568-2597. [4] Ding WY (1986) On a conformally invariant elliptic equation on $\mathbb{R}.n$. Commun Math Phys 107: 331-335. [5] Fernández JC, Petean J (2020) Low energy nodal solutions to the Yamabe equation. J Differ Equations 268: 6576-6597. [6] Grossi M, Pacella F (1990) Positive solutions of nonlinear elliptic equations with critical Sobolev exponent and mixed boundary conditions. P Roy Soc Edinb A 116: 23-43. [7] Lions PL, Pacella F (1990) Isoperimetric inequalities for convex cones. P Am Math Soc 109: 477- 485. [8] Lions PL, Pacella F, Tricarico M (1988) Best constants in Sobolev inequalities for functions vanishing on some part of the boundary and related questions. Indiana U Math J 37: 301-324. [9] Weth T (2006) Energy bounds for entire nodal solutions of autonomous superlinear equations. Calc Var Partial Dif 27: 421-437. [10] Willem M (1996) Minimax Theorems, Boston: Birkh?user Boston.
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