Nonexistence of positive solutions for a system of integral equations on$R^n_+$ and applications

Dongyan Li; Yongzhong Wang

doi:10.3934/cpaa.2013.12.2601

Article Contents

2013, Volume 12, Issue 6: 2601-2613. Doi: 10.3934/cpaa.2013.12.2601

This issue Previous Article Four positive solutions of a quasilinear elliptic equation in $ R^N$ Next Article A stability result for the Stokes-Boussinesq equations in infinite 3d channels

Nonexistence of positive solutions for a system of integral equations on $R^n_+$ and applications

Dongyan Li^1, and
Yongzhong Wang^1,

1.
Department of Applied Mathematics, Key Laboratory of Space Applied Physics and Chemistry, Ministry of Education, Northwestern Polytechnical University, Xi'an, Shaanxi 710129

Received: June 30, 2012

Revised: December 31, 2012

Published: October 2013

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Abstract

Let $R^n_+$ be the $n$-dimensional upper half Euclidean space, $m$ be a positive integer. In this paper, we consider the following system of integral equations on $R^n_+$: \begin{eqnarray} u(x)=\int_{R^n_+}G(x,y)v^q(y)dy, \\ v(x)=\int_{R^n_+}G(x,y)u^p(y)dy, \end{eqnarray} where \begin{eqnarray} G(x,y)=\frac{c_n}{|x-y|^{n-2m}}\int_0^{\frac{4x_ny_n}{|x-y|^2}}\frac{z^{m-1}}{(z+1)^{\frac{n}{2}}}dz \end{eqnarray} with $0 < 2m < n$ and $p,q>1$. Nonexistence of positive solution is proved by using the method of moving planes in integral forms. We also obtain the equivalence between the system of integral equations and corresponding partial differential equations.

Keywords:

System of integral equations,
nonexistence,
method of moving planes in integral forms,
equivalence,
a priori bound.

Mathematics Subject Classification: Primary: 35B45, 35B53; Secondary: 35J91.

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