Article Contents
Article Contents

# On the integral systems with negative exponents

• 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.

 Citation:

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