A note on the monotonicity of solutions for fractional equations in half-spaces

Barrios, B.; García-Melián, J.; Quaas, A.

doi:10.1090/proc/14469

A note on the monotonicity of solutions for fractional equations in half-spaces
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by B. Barrios, J. García-Melián and A. Quaas PDF

Proc. Amer. Math. Soc. 147 (2019), 3011-3019 Request permission

Abstract:

In this work we consider the nonlocal elliptic problem \begin{equation*} \left \{ \begin {array}{ll} (-\Delta )^s u = f(u) & \text {in } \mathbb {R}^N_+,\\[4pt] \ \ u=0 & \text {in } \mathbb {R}^N \setminus \mathbb {R}^N_+, \end{array} \right . \end{equation*} where $(-\Delta )^s$, $0<s<1$, stands for the fractional Laplacian and $\mathbb {R}^N_+=\{(x’,x_N)\in \mathbb {R}^N:\ x_N>0\}$ is the half-space. It is shown that nonnegative, nontrivial, bounded, classical solutions of this problem are positive and strictly monotone in the $x_N$ direction, assuming only that $f$ is locally Lipschitz, thereby improving a previous result of the authors which required $f$ to be $C^1$.

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