Some symmetry results for integral equations involving Wolff potential on bounded domains (Q450599)

This paper is devoted to an investigation of the following systems of nonlinear integral equations involving the Wolff potential on a bounded domain NEWLINE\[NEWLINE u(x)=\int\limits_{0}^{\infty}\left[\frac{\int_{B_t(x)\cap\Omega}u^{a}(y)v^b(y)dy}{t^{n-\alpha\gamma}}\right]^{\frac{1}{\gamma-1}}\frac{dt}{t}, \quad x\in\Omega;NEWLINE\]NEWLINE NEWLINE\[NEWLINE v(x)=\int\limits_{0}^{\infty}\left[\frac{\int_{B_t(x)\cap\Omega}u^{c}(y)v^d(y)dy}{t^{n-\beta\kappa}}\right]^{\frac{1}{\gamma-1}}\frac{dt}{t}, \quad x\in\Omega,NEWLINE\]NEWLINE where \(\alpha,\beta,\gamma,\kappa,a,b,c,d\) are constants. It is shown that \(u\) and \(v\) are constants on \(\partial\Omega\) if and only if \(\Omega\) is a ball. The symmetry results for such systems are also obtained.