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

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  • 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 $B_1(0)$.
    Mathematics Subject Classification: Primary: 35J47, 35B06, 35B53; Secondary: 35B65.

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