@ARTICLE{10.21494/ISTE.OP.2021.0646, 

TITLE={A priori estimates for super-linear elliptic equation: the Neumann boundary value problem}, 
  
AUTHOR={Abdellaziz Harrabi, Belgacem Rahal, Abdelbaki Selmi, }, 
	
JOURNAL={Advances in Pure and Applied Mathematics}, 
	
VOLUME={12}, 

NUMBER={Issue 2 (May 2021)}, 
	
YEAR={2021}, 

URL={https://openscience.fr/A-priori-estimates-for-super-linear-elliptic-equation-the-Neumann-boundary}, 
	
DOI={10.21494/ISTE.OP.2021.0646}, 
	
ISSN={1869-6090}, 
	
ABSTRACT={$$$\mbox{In this paper we study the nonexistence of finite Morse index solutions of the following} \\\mbox{Neumann boundary value problems}\\
{(Eq.H)} \begin{cases}  -\Delta u = (u^{+})^{p} \;\; \text{in $ \mathbb{R}_+^N$}, \\ \frac{\partial u}{\partial x_{N}}=0 \quad\quad\;\; \text{ on  $ \partial\mathbb{R}_+^N$}, \\ u  \in C^2(\overline{\mathbb{R}_+^N}) \mbox{  and sign-changing, }\\u^+ \mbox{  is bounded and } i(u)<\infty,\end{cases}\\
\mbox{or}\\
{(Eq.H')}\begin{cases}-\Delta u = |u|^{p-1}u  \;\; \text{in  $ \mathbb{R}_+^N$}, \\ \frac{\partial u}{\partial x_{N}}=0 \;\;\;\;\;\;\;\;\quad\text{ on  $ \partial\mathbb{R}_+^N$}, \\ u   \in C^2(\overline{\mathbb{R}_+^N}),\\ u \mbox{  is bounded and }  i(u) < \infty.\end{cases}\\
\mbox{  As a consequence, we establish the relevant Bahri-Lions's }L^\infty\mbox{-estimate [3] via}\\ \mbox{the boundedness of Morse index of solutions to}\\
\begin{equation}\label{1.1}
 \left\{\begin{array}{lll} -\Delta u=f(x,u) &\text{in  $ \Omega,$}\\
  \frac{\partial u}{\partial \nu}=0 &\text{on $\partial \Omega,$}
\end{array}
\right.
\end{equation}\\
\mbox{where} f \mbox{ has an asymptotical behavior at in-nity} \mbox{which is not necessarily the same at}  \pm\infty. \\\mbox{Our results complete previous Liouville type theorems and } L^\infty\mbox{-bounds via Morse index} \\\mbox{ obtained in [3, 6, 13, 16, 12, 21].}$$$}}