Titre : Estimations à priori pour l’équation elliptique super-linéaire : le problème de la valeur au bord de Neumann 

Auteurs : Abdellaziz Harrabi, Belgacem Rahal, Abdelbaki Selmi,  

Revue : Avancées en Mathématiques Pures et Appliquées 

Numéro : Numéro 2 (Mai 2021) 

Volume : 12 

Date : 2021/04/26 

DOI : 10.21494/ISTE.OP.2021.0646 

ISSN : 1869-6090 

Résumé : $$$\mbox{In this paper we study the nonexistence of finite Morse index solutions of the following Neumann boundary value problems}\\
{(Eq.H)} \begin{cases}  -\Delta u = (u^{+})^{p} \;\; \text{in $ \mathbb{R}_+^N$}, \\ \frac{\partial u}{\partial x_{N}}=0 \quad\quad\;\; \text{ on  $ \partial\mathbb{R}_+^N$}, \\ u  \in C^2(\overline{\mathbb{R}_+^N}) \mbox{  and sign-changing, }\\u^+ \mbox{  is bounded and } i(u)<\infty,\end{cases}\\
\mbox{or}\\
{(Eq.H')}\begin{cases}-\Delta u = |u|^{p-1}u \text{ in $ \mathbb{R}_+^N$}, \\ \frac{\partial u}{\partial x_{N}}=0 \;\;\;\;\;\;\;\;\quad\text{ on  $ \partial\mathbb{R}_+^N$}, \\ u   \in C^2(\overline{\mathbb{R}_+^N}),\\ u \mbox{  is bounded and }  i(u) < \infty.\end{cases}\\
\mbox{  As a consequence, we establish the relevant Bahri-Lions's }L^\infty\mbox{-estimate [3] via the boundedness of Morse index of solutions to}\\
\begin{equation}\label{1.1}
 \left\{\begin{array}{lll} -\Delta u=f(x,u) &\text{in  $ \Omega,$}\\
  \frac{\partial u}{\partial \nu}=0 &\text{on $\partial \Omega,$}
\end{array}
\right.
\end{equation}\\
\mbox{where} f \mbox{ has an asymptotical behavior at in-nity} \mbox{which is not necessarily the same at}  \pm\infty. \mbox{Our results complete previous Liouville}\\ \mbox{ type theorems and } L^\infty\mbox{-bounds via Morse index obtained in [3, 6, 13, 16, 12, 21].}$$$ 

Éditeur : ISTE OpenScience