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Zhang, Rong

On the Wolff-type integral system with negative exponents. (English) Zbl 1505.45005

Rocky Mt. J. Math. 52, No. 6, 2211-2228 (2022).
Summary: Here, we are concerned with the positive continuous entire solutions of the Wolff-type integral system \[ \begin{cases} u(x)=C_{1} (x) W_{\beta,\gamma} (v^{-q}) (x), & \qquad u, v>0 \text{ in }\mathbb{R}^n, \\ v(x)=C_{2} (x) W_{\beta,\gamma} (u^{-p}) (x), & \qquad p, q>0, \end{cases} \] where \(n\geq 1\), \(\gamma>1\), \(\beta>0\) and \(\beta\gamma\neq n\). In addition, \(C_i(x)\) \((i=1, 2)\) are some double bounded functions. When \(\beta\gamma\in (0, n)\), the Serrin-type condition is critical for existence of positive solutions for some double bounded functions \(C_i(x)\) (\(i=1, 2\)). Such an integral equation system is related to the study of the \(\gamma\)-Laplace system and \(k\)-Hessian system with negative exponents. Estimated by the integral of the Wolff type potential, we obtain the asymptotic rates and the integrability of positive solutions, and study whether the radial solutions exist.

MSC:

45G15 Systems of nonlinear integral equations
45M05 Asymptotics of solutions to integral equations
45M20 Positive solutions of integral equations

Keywords:

Wolff-type potential; Serrin-type condition; \(\gamma\)-Laplace system; \(k\)-Hessian system; asymptotic limit
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References:

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