Monotonicity of solutions for some nonlocal elliptic problems in half-spaces

Autor: Begoña Barrios, L. Del Pezzo, Jorge García-Melián, Alexander Quaas
Rok vydání: 2017
Předmět:
Zdroj: CONICET Digital (CONICET)
Consejo Nacional de Investigaciones Científicas y Técnicas
instacron:CONICET
ISSN: 1432-0835
0944-2669
DOI: 10.1007/s00526-017-1133-9
Popis: In this paper we consider classical solutions $u$ of the semilinear fractional problem $(-\Delta)^s u = f(u)$ in $\mathbb{R}^N_+$ with $u=0$ in $\mathbb{R}^N \setminus \mathbb{R}^N_+$, where $(-\Delta)^s$, $00\}$ is the half-space and $f\in C^1$ is a given function. With no additional restriction on the function $f$, we show that bounded, nonnegative, nontrivial classical solutions are indeed positive in $\mathbb{R}^N_+$ and verify $$ \frac{\partial u}{\partial x_N}>0 \quad \hbox{in } \mathbb{R}^N_+. $$ This is in contrast with previously known results for the local case $s=1$, where nonnegative solutions which are not positive do exist and the monotonicity property above is not known to hold in general even for positive solutions when $f(0)
Comment: 18 pages
Databáze: OpenAIRE
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