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

## A priori estimates for super-linear elliptic equation: the Neumann boundary value problem

Abdellaziz Harrabi
Northern Border university
Saudi Arabia

Belgacem Rahal
Institut Supérieur des Sciences Appliquées et de Technologie de Kairouan
Tunisia

Abdelbaki Selmi
Northern Border university
Saudi Arabia

Publié le 26 avril 2021   DOI : 10.21494/ISTE.OP.2021.0646

### Keywords

$\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}\\ $$\label{1.1} \left\{\begin{array}{lll} -\Delta u=f(x,u) &\text{in \Omega,}\\ \frac{\partial u}{\partial \nu}=0 &\text{on \partial \Omega,} \end{array} \right.$$\\ \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].}$

$\mbox{In this paper we study the nonexistence of finite Morse index solutions of the following} \\\mbox{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}\\ \mbox{the boundedness of Morse index of solutions to}\\ $$\label{1.1} \left\{\begin{array}{lll} -\Delta u=f(x,u) &\text{in \Omega,}\\ \frac{\partial u}{\partial \nu}=0 &\text{on \partial \Omega,} \end{array} \right.$$\\ \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 type theorems and } L^\infty\mbox{-bounds via Morse index} \\\mbox{ obtained in [3, 6, 13, 16, 12, 21].}$