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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).

MSC:

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