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On the integral systems with negative exponents

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  • This paper is concerned with the integral system $$\left \{ \begin{array}{ll} &u(x)=\int_{R^n}\frac{|x-y|^\lambda dy}{v^q(y)},\quad u>0~in~R^n,\\ &v(x)=\int_{R^n}\frac{|x-y|^\lambda dy}{u^p(y)},\quad v>0~in~R^n, \end{array} \right. $$ where $n \geq 1$, $p,q,\lambda \neq 0$. Such an integral system appears in the study of the conformal geometry. We obtain several necessary conditions for the existence of the $C^1$ positive entire solutions, particularly including the critical condition $$ \frac{1}{p-1}+\frac{1}{q-1}=\frac{\lambda}{n}, $$ which is the necessary and sufficient condition for the invariant of the system and some energy functionals under the scaling transformation. The necessary condition $\frac{1}{p-1}+\frac{1}{q-1}=\frac{\lambda}{n}$ can be relaxed to another weaker one $\min\{p,q\}>\frac{n+\lambda}{\lambda}$ for the system with double bounded coefficients. In addition, we classify the radial solutions in the case of $p=q$ as the form $$ u(x)=v(x)=a(b^2+|x-x_0|^2)^{\frac{\lambda}{2}} $$ with $a,b>0$ and $x_0 \in R^n$. Finally, we also deduce some analogous necessary conditions of existence for the weighted system.
    Mathematics Subject Classification: 45E10, 45G15, 45M05, 45M20.


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