## 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 $$0$$.

### MSC:

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

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