scholarly journals Some properties of positive solutions for an integral system with the double weighted Riesz potentials

2016 ◽  
Vol 15 (6) ◽  
pp. 2117-2134
Author(s):  
Jiankai Xu ◽  
Song Jiang ◽  
Huoxiong Wu
Keyword(s):  
Positive Solutions ◽  
Riesz Potentials ◽  
Integral System

scholarly journals On the Wolff-type Integral System with Negative Exponents

2021 ◽  
Author(s):  
Rong Zhang ◽  
Ling Li
Keyword(s):  
Integral Equation ◽  
Positive Solutions ◽  
Equation System ◽  
Type Condition ◽  
Radial Solutions ◽  
Entire Solutions ◽  
Integral System ◽  
Type Integral ◽  
Bounded Functions ◽  
Type Potential

In this paper, we are concerned with the positive continuous entire solutions of the Wolff-type integral system \begin{equation*} \left\{ \begin{array}{ll} &u(x) =C_{1}(x)W_{\beta,\gamma} (v^{-q})(x), \\[3mm] &v(x) =C_{2}(x)W_{\beta,\gamma} (u^{-p})(x), \end{array} \right. \end{equation*} where $n\geq1$, $\min\{p,q\}>0$, $\gamma>1$, $\beta>0$ and $\beta\gamma\neq n$. In addition, $C_{i}(x) \ (i=1,2)$ are some double bounded functions. If $\beta\gamma\in (0,n)$, the Serrin-type condition is critical for existence of the 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 studied whether the radial solutions exist.

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