## 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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### References:

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