Nonlinear elliptic systems involving the fractional Laplacian in the unit ball and on a half space

Chenchen  Mou

doi:10.3934/cpaa.2015.14.2335

Article Contents

2015, Volume 14, Issue 6: 2335-2362. Doi: 10.3934/cpaa.2015.14.2335

This issue Previous Article Krasnosel'skii type formula and translation along trajectories method on the scale of fractional spaces Next Article Harnack inequality for degenerate elliptic equations and sum operators

Nonlinear elliptic systems involving the fractional Laplacian in the unit ball and on a half space

Chenchen Mou^1,

1.
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA

Received: December 31, 2014

Revised: May 31, 2015

Published: October 2015

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Abstract

In this paper, we study the following nonlinear elliptic system \begin{eqnarray} \left\{\begin{array}{ll} (-\Delta)^{\frac{\alpha}{2}}u_i=f_i(u),\ x\in \Omega,\quad i=1,...,m, \\ u_i(x)=0, \quad \quad\quad \ \ x\in \Omega^c,\quad i=1,...,m, \end{array} \right. \end{eqnarray} where $0 < \alpha < 2$ and $\Omega$ is either the unit ball $B_1(0)=\{x\in \mathbb R^n | \|x\| < 1 \}$ or the half space $\mathbb R_+^n = \{x=(x_1,...,x_n)\in \mathbb R^n | x_n > 0\}$. Instead of investigating the pseudo differential system directly, we study an equivalent integral system, i.e., \begin{eqnarray} u_i(x)=\int_{B_1(0)}G_1(x,y)f_i(u(y))dy,\quad x\in B_1(0),\quad i=1,...,m, \end{eqnarray} and \begin{eqnarray} u_i(x)=C_ix_n^{\frac{\alpha}{2}}+\int_{\mathbb{R}_+^n}G_{\infty}(x,y)f_i(u(y))dy,\quad x\in \mathbb{R}_+^n,\quad i=1,...,m, \end{eqnarray} where $C_i$ are non-negative constants, $G_1(x,y)$ is Green's function for $B_1(0)$ and $G_{\infty}(x,y)$ is Green function of $\mathbb R_+^n$. We use the method of moving planes in integral forms to prove the radial symmetry and monotonicity of positive solutions in $B_1(0)$ and non-existence of positive solutions in $\mathbb R_+^n$. Moreover, we also study regularity of positive solutions in $B_1(0)$.

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Mathematics Subject Classification: Primary: 35J47, 35B06, 35B53; Secondary: 35B65.

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