Journal of Inequalities and Applications (Oct 2016)
Liouville type theorems for the system of fractional nonlinear equations in R + n ${R^{n}_{+}}$
Abstract
Abstract In this paper we consider the following system of fractional nonlinear equations in the half space R + n ${R^{n}_{+}}$ : 1 { ( − Δ ) α 2 u 1 ( x ) = x n γ u 1 α 1 ( x ) u 2 β 1 ( x ) , x ∈ R + n , ( − Δ ) α 2 u 2 ( x ) = x n γ u 1 α 2 ( x ) u 2 β 2 ( x ) , x ∈ R + n , u 1 ( x ) = u 2 ( x ) = 0 , x ∉ R + n , $$ \textstyle\begin{cases} (-\Delta)^{\frac{\alpha}{2}}u_{1}(x)=x^{\gamma}_{n}u_{1}^{\alpha _{1}}(x)u_{2}^{\beta_{1}}(x), &\mbox{$ x \in R^{n}_{+}$},\\ (-\Delta)^{\frac{\alpha}{2}}u_{2}(x)=x^{\gamma}_{n}u_{1}^{\alpha _{2}}(x)u_{2}^{\beta_{2}}(x), &\mbox{$x \in R^{n}_{+}$},\\ u_{1}(x)=u_{2}(x)=0, &\mbox{$x \notin R^{n}_{+}$}, \end{cases} $$ where γ ≥ 0 $\gamma\geq0$ , 0 0 $\beta_{i} >0$ , i = 1 , 2 $i=1,2$ . First, we use the Kelvin transform and the method of moving planes in integral forms to prove that (1) is equivalent to the following system of integral equations with 1 < α i + β i ≤ n + α + 2 γ n − α $1<\alpha_{i}+\beta_{i}\leq\frac{n+\alpha +2\gamma}{n-\alpha}$ : 2 { u 1 ( x ) = ∫ R + n G ( x , y ) y n γ u 1 α 1 ( y ) u 2 β 1 ( y ) d y , x ∈ R + n , u 2 ( x ) = ∫ R + n G ( x , y ) y n γ u 1 α 2 ( y ) u 2 β 2 ( y ) d y , x ∈ R + n , $$ \textstyle\begin{cases} u_{1}(x)=\int_{R^{n}_{+}}G(x,y)y^{\gamma}_{n}u_{1}^{\alpha _{1}}(y)u_{2}^{\beta_{1}}(y)\,dy, &\mbox{$x \in R^{n}_{+}$},\\ u_{2}(x)=\int_{R^{n}_{+}}G(x,y)y^{\gamma}_{n}u_{1}^{\alpha _{2}}(y)u_{2}^{\beta_{2}}(y)\,dy, &\mbox{$x \in R^{n}_{+}$}, \end{cases} $$ where G ( x , y ) $G(x,y)$ is the Green’s function associated with ( − Δ ) α 2 $(-\Delta )^{\frac{\alpha}{2}}$ in R + n $R^{n}_{+}$ . Then we continue work on integral systems (2) to establish Liouville type theorems, i.e. the nonexistence of positive solutions in the subcritical case and the critical case, 1 < α i + β i ≤ n + α + 2 γ n − α $1<\alpha_{i}+\beta_{i}\leq\frac{n+\alpha+2\gamma}{n-\alpha}$ .
Keywords
