Higher and fractional order Hardy-Sobolev type equations (Q2928532)

In this paper the authors are concerned with the higher order fractional Hardy-Sobolev equation NEWLINE\[NEWLINE (-\Delta )^{\alpha/2}u(x)=\frac{u^p(x)}{|y|^t},\quad x=(y,z)\in (\mathbb R^{k}\setminus\{0\})\times \mathbb R^{n-k}\qquad(1) NEWLINE\]NEWLINE where \(0<\alpha<n\), \(0<t<\min\{\alpha, k\}\) and \(1<p<\frac{n+\alpha-2t}{n-\alpha}\).NEWLINENEWLINEFor even values of \(\alpha\) the equivalence between the above equation and the integral equation NEWLINE\[NEWLINE u(x)=\int_{\mathbb R^n}G(x,\xi)\frac{u^p(x)}{|y|^t}d\xi NEWLINE\]NEWLINE is established, where \(G\) is the Green's function of \((-\Delta)^{\alpha/2}\) in \(\mathbb R^n\). Using the method of moving planes, the authors show that each non-negative solution of equation (1) is radially symmetric about some point \(z_0\in \mathbb R^{n-k}\) and radially symmetric and monotone decreasing about the origin in \(\mathbb R^k\). Some nonexistence results are also obtained in the subcritical regime.