Qualitative properties of solutions to an integral system associated with the Bessel potential

Lu  Chen; Zhao  Liu; Guozhen Lu

doi:10.3934/cpaa.2016.15.893

Article Contents

2016, Volume 15, Issue 3: 893-906. Doi: 10.3934/cpaa.2016.15.893

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Qualitative properties of solutions to an integral system associated with the Bessel potential

1.
School of Mathematical Sciences, Beijing Normal University, Beijing 100875
2.
Department of Mathematics, Wayne State University, Detroit, MI 48202

Received: July 31, 2015

Revised: October 31, 2015

Published: April 2016

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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.

Mathematics Subject Classification: Primary: 35J48; Secondary: 35B06, 45G15.

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