Hardy-Sobolev critical elliptic equations with boundary singularities. (English) Zbl 1232.35064

Summary: Unlike the non-singular case \(s=0\), or the case when \(0\) belongs to the interior of a domain \(\Omega\) in \(\mathbb R^n\) \((n\geq 3)\), we show that the value and the attainability of the best Hardy-Sobolev constant on a smooth domain \(\Omega\),
\[ \mu_s(\Omega):=\inf\left\{\int_{\Omega}|\nabla u|^2\,dx; u\in H^1_0(\Omega) \text{ and }\;\int_\Omega\frac{|u|^{2^*(s)}}{|x|^s}=1\right\}, \]
when \(0<s<2\), \(2^{*}(s):= \frac{2(n-s)}{(n-2)}\), and when 0 is on the boundary \(\partial\Omega\) are closely related to the properties of the curvature of \(\partial\Omega\) at \(0\). These conditions on the curvature are also relevant to the study of elliptic partial differential equations with singular potentials of the form:
\[ -\Delta u=\frac{|u|^{2^{*}(s)-2} u}{|x|^s} +f(x,u)\quad \text{ in } \Omega, \]
where \(f\) is a lower order perturbative term at infinity and \(f(x,0)=0\). We show that the positivity of the sectional curvature at \(0\) is relevant when dealing with Dirichlet boundary conditions, while the Neumann problems seem to require the positivity of the mean curvature at \(0\).


35J61 Semilinear elliptic equations
35B33 Critical exponents in context of PDEs
35J20 Variational methods for second-order elliptic equations
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