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The properties of positive solutions to an integral system involving Wolff potential. (English) Zbl 1277.45003

Summary: We consider the positive solutions of the following weighted integral system involving Wolff potential in \(\mathbb{R}^{n}\): \[ \begin{cases} u(x) = R_1(x)W_{\beta,\gamma}\left(\frac{v^q}{|y|^{\sigma}}\right)(x), \\ v(x) = R_2(x)W_{\beta,\gamma}\left(\frac{u^p}{|y|^{\sigma}}\right)(x).\end{cases} \tag{0.1} \] This system is helpful to understand some nonlinear PDEs and other nonlinear problems. Different from the case of \(\sigma=0\), it is difficult to handle the properties of the solutions since there is singularity at origin. First, we overcome this difficulty by modifying and refining the new method which was introduced to explore the integrability result established by Ch. Ma et al. [Adv. Math. 226, No. 3, 2676–2699 (2011; Zbl 1209.45006)], and obtain an optimal integrability. Second, we use the method of moving planes to prove the radial symmetry for the positive solutions of (0.1) when \(R_{1}(x) \equiv R_{2}(x) \equiv 1\). Based on these results, by intricate analytical techniques, we obtain the estimate of the decay rates of those solutions near infinity.

MSC:

45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
45G05 Singular nonlinear integral equations
45M05 Asymptotics of solutions to integral equations
45M20 Positive solutions of integral equations

Citations:

Zbl 1209.45006
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References:

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