Quantitative analysis of some system of integral equations. (English) Zbl 1113.45006

The optimization problem of finding optimal constants \(C\) for the double weighted Hardy-Littlewood-Sobolev inequality
\[ \int_{{\mathbb R}^n}\int_{{\mathbb R}^n}{f(x)g(y)\over | x| ^\alpha| x-y| ^\lambda| y| ^\beta}\,dx\,dy\leq C\| f\|_\rho\| g\|_\sigma \]
(with \(\alpha,\beta,\lambda,\rho,\sigma\) satisfying well-known relations) leads to Euler-Lagrange equations which can be transformed to the system \[ u(x)=\int_{{\mathbb R}^n}{v(y)^q\over | x| ^\alpha| x-y| ^\lambda| y| ^\beta}\,dy, \]
\[ v(x)=\int_{{\mathbb R}^n}{u(y)^p\over | x| ^\beta| x-y| ^\lambda| y| ^\alpha}\,dy. \] For nonnegative solutions of this equation optimal powers \(r,s\) with \(u\in L^r\), \(v\in L^s\) are calculated in case \(\alpha,\beta\geq0\) and partially also in case \(\alpha\beta<0\). Analogous results hold if the above kernels are multiplied with functions which are bounded from above (for the estimates) resp. from \(0\) (for the optimality).


45G15 Systems of nonlinear integral equations
26D15 Inequalities for sums, series and integrals
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