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A singularity analysis of positive solutions to an Euler-Lagrange integral system. (English) Zbl 1220.45004

The authors deal with the following system of Euler-Lagrange equations in \(\mathbb R^n\): \[ u(x)=\frac{1}{|x|^{\alpha}}\int_{\mathbb R^n} \frac{v(y)^q}{|y|^{\beta}|x-y|^{\lambda}}dy,\qquad v(x)=\frac{1}{|x|^{\beta}}\int_{\mathbb R^n} \frac{u(y)^p}{|y|^{\alpha}|x-y|^{\lambda}}dy \] which appears in a natural way if one wants to obtain the best constants in the well-known weighted Hardy-Littlewood-Sobolev inequality. They deal with the asymptotic behaviour of positive solutions to the above system and obtain some asymptotic estimates in the case when \(\alpha+\beta\geq 0\). These results are extensions of previous ones proved by Y. Lei and C. Ma [Commun. Pure Appl. Anal. 10, 193–207 (2011)], for the case \(\alpha,\beta\geq 0\).

MSC:

45G15 Systems of nonlinear integral equations
45G05 Singular nonlinear integral equations
45M20 Positive solutions of integral equations
26D10 Inequalities involving derivatives and differential and integral operators
45M05 Asymptotics of solutions to integral equations
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