Classification of positive solutions to an integral system with the poly-harmonic extension operator

2018 ◽  
Vol 61 (9) ◽  
pp. 1603-1616 ◽  
Author(s):  
Sufang Tang ◽  
Jingbo Dou
Keyword(s):  
Positive Solutions ◽  
Extension Operator ◽  
Harmonic Extension ◽  
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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