Regularity and nonexistence of solutions for a system involving the fractional Laplacian

De  Tang; Yanqin  Fang

doi:10.3934/cpaa.2015.14.2431

Article Contents

2015, Volume 14, Issue 6: 2431-2451. Doi: 10.3934/cpaa.2015.14.2431

This issue Previous Article Least energy solutions for an elliptic problem involving sublinear term and peaking phenomenon Next Article Wellposedness of the Keller-Segel Navier-Stokes equations in the critical Besov spaces

Regularity and nonexistence of solutions for a system involving the fractional Laplacian

De Tang^1, and
Yanqin Fang^1,

1.
School of Mathematics, Hunan University, Changsha, 410082, China

Received: February 28, 2015

Revised: June 30, 2015

Published: October 2015

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Abstract

We consider a system involving the fractional Laplacian \begin{eqnarray} \left\{ \begin{array}{ll} (-\Delta)^{\alpha_{1}/2}u=u^{p_{1}}v^{q_{1}} & \mbox{in}\ \mathbb{R}^N_+,\\ (-\Delta)^{\alpha_{2}/2}v=u^{p_{2}}v^{q_{2}} &\mbox{in}\ \mathbb{R}^N_+,\\ u=v=0,&\mbox{in}\ \mathbb{R}^N\backslash\mathbb{R}^N_+, \end{array} \right. \end{eqnarray} where $\alpha_{i}\in (0,2)$, $p_{i},q_{i}>0$, $i=1,2$. Based on the uniqueness of $\alpha$-harmonic function [9] on half space, the equivalence between (1) and integral equations \begin{eqnarray} \left\{ \begin{array}{ll} u(x)=C_{1}x_{N}^{\frac{\alpha_{1}}{2}}+\displaystyle\int_{\mathbb{R}_{+}^{N}}G^{1}_{\infty}(x,y)u^{p_{1}}(y)v^{q_{1}}(y)dy,\\ v(x)=C_{2}x_{N}^{\frac{\alpha_{2}}{2}}+\displaystyle\int_{\mathbb{R}_{+}^{N}}G^{2}_{\infty}(x,y)u^{p_{2}}(y)v^{q_{2}}(y)dy. \end{array} \right. \end{eqnarray} are derived. Based on this result we deal with integral equations (2) instead of (1) and obtain the regularity. Especially, by the method of moving planes in integral forms which is established by Chen-Li-Ou [12], we obtain the nonexistence of positive solutions of integral equations (2) under only local integrability assumptions.

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Mathematics Subject Classification: Primary: 35R11; Secondary: 35A01, 35B53.

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