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

Abstract

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 $(-\Delta {)}^s$, $0<s<1$, stands for the fractional laplacian, $N\ge 2$, ${\mathbb{R}}_{+}^N=\{x=(x^{\prime },x_N)\in {\mathbb{R}}^N:x_N>0\}$ 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 $$\frac{\partial u}{\partial x_N}>0in{\mathbb{R}}_{+}^N.$$ 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 $f(0)<0$.

18 pages

References in corpus (1)

• On fractional elliptic equations in Lipschitz sets and epigraphs: regularity, monotonicity and rigidity results

Cited by in corpus (3)

• On fractional elliptic equations in Lipschitz sets and epigraphs: regularity, monotonicity and rigidity results
• Maximum principles and the method of moving planes for the uniformly elliptic nonlocal Bellman operator and applications
• Semilinear integro-differential equations, II: one-dimensional and saddle-shaped solutions to the Allen-Cahn equation