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

Abstract

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

Keywords

• hardy-sobolev inequality
• positive minimizers
• parametrized curve
• mass
• green function