Qualitative properties of solutions to an integral system associated with the Bessel potential
Lu Chen  School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China (email) Abstract: In this paper, we study a differential system associated with the Bessel potential: \begin{eqnarray}\begin{cases} (I\Delta)^{\frac{\alpha}{2}}u(x)=f_1(u(x),v(x)),\\ (I\Delta)^{\frac{\alpha}{2}}v(x)=f_2(u(x),v(x)), \end{cases}\end{eqnarray} where $f_1(u(x),v(x))=\lambda_1u^{p_1}(x)+\mu_1v^{q_1}(x)+\gamma_1u^{\alpha_1}(x)v^{\beta_1}(x)$, $f_2(u(x),v(x))=\lambda_2u^{p_2}(x)+\mu_2v^{q_2}(x)+\gamma_2u^{\alpha_2}(x)v^{\beta_2}(x)$, $I$ is the identity operator and $\Delta=\sum_{j=1}^{n}\frac{\partial^2}{\partial x^2_j}$ is the Laplacian operator in $\mathbb{R}^n$. Under some appropriate conditions, this differential system is equivalent to an integral system of the Bessel potential type. By the regularity lifting method developed in [4] and [18], we obtain the regularity of solutions to the integral system. We then apply the moving planes method to obtain radial symmetry and monotonicity of positive solutions. We also establish the uniqueness theorem for radially symmetric solutions. Our nonlinear terms $f_1(u(x), v(x))$ and $f_2(u(x), v(x))$ are quite general and our results extend the earlier ones even in the case of single equation substantially.
Keywords: Bessel potential, method of moving planes in integral forms, radial symmetry, regularity, uniqueness.
Received: August 2015; Revised: November 2015; Available Online: February 2016. 
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