Abstract
Résumé
Keywords
Mots-clés
In this paper we study the nonexistence of finite Morse index solutions of the followingNeumann boundary value problems(Eq.H)⎧⎪
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⎪⎨⎪
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⎪⎩−Δu=(u+)pin RN+,∂u∂xN=0 on ∂RN+,u∈C2(¯¯¯¯¯¯¯¯RN+) and sign-changing, u+ is bounded and i(u)<∞,or(Eq.H′)⎧⎪
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⎪⎨⎪
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⎪⎩−Δu=|u|p−1uin RN+,∂u∂xN=0 on ∂RN+,u∈C2(¯¯¯¯¯¯¯¯RN+),u is bounded and i(u)<∞. As a consequence, we establish the relevant Bahri-Lions's L∞-estimate [3] viathe boundedness of Morse index of solutions to{−Δu=f(x,u)in Ω,∂u∂ν=0on ∂Ω,wheref has an asymptotical behavior at in-nitywhich is not necessarily the same at±∞.Our results complete previous Liouville type theorems and L∞-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 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.