Influence of an L^{p}-perturbation on Hardy-Sobolev inequality with singularity a curve

We consider a bounded domain \(\Omega\) of \(\mathbb{R}^N\), \(N \geq 3\), \(h\) and \(b\) continuous functions on \(\Omega\). Let \(\Gamma\) be closed curve contained in study existence positive solutions \(u \in H^1_0(\Omega)\) to the perturbed Hardy-Sobolev equation: \[-\Delta u+hu+bu^{1+\delta}=\rho^{-\sigma}_{\Gamma} u^{2^*_{\sigma}-1} \quad \textrm{ } \Omega,\] where \(2^*_{\sigma}:=\frac{2(N-\sigma)}{N-2}\) is critical exponent, \(\sigma\in [0,2)\), \(0\lt\delta\lt\frac{4}{N-2}\) \(\rho_{\Gamma}\) distance function \(\Gamma\). show that minimizers does not depend local geometry nor potential \(h\). For \(N=3\), ground-state solution may depends trace regular part Green \(-\Delta+h\) or \(b\). This due perturbative term order \(1+\delta\).