Optimal integrability of some system of integral equations

Abstract

We obtain the optimal integrability for positive solutions of the Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality in {ℝ}^n:

\{\begin{array}{l}u(x)={\displaystyle \frac{\mathrm{1}}{{\left|x\right|}^{\alpha }}}{\int }_{{ℝ}^n}{\displaystyle \frac{\upsilon {(y)}^q}{{\left|y\right|}^{\beta }{\left|x-y\right|}^{\lambda }}}dy,\\ u(x)={\displaystyle \frac{\mathrm{1}}{{\left|x\right|}^{\beta }}}{\int }_{{ℝ}^n}{\displaystyle \frac{\upsilon {(y)}^p}{{\left|y\right|}^{\alpha }{\left|x-y\right|}^{\lambda }}}dy.\end{array}

C. Jin and C. Li [Calc. Var. Partial Differential Equations, 2006, 26: 447-457] developed some very interesting method for regularity lifting and obtained the optimal integrability for p, q>1. Here, based on some new observations, we overcome the difficulty there, and derive the optimal integrability for the case of p, q≥1 and pq≠ 1. This integrability plays a key role in estimating the asymptotic behavior of positive solutions when |x| → 0 and when |x| → ∞.

Key words： Integral equation; weighted Hardy-Littlewood-Sobolev inequality; integrability interval

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