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

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

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

Morse index Neumann boundary value problem supercritical growth Liouville-type problems L∞1 -bounds.

Morse index Neumann boundary value problem supercritical growth Liouville-type problems L∞1 -bounds.