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Hardy-Sobolev critical equation with boundary singularity: multiplicity and stability of the Pohozaev obstruction. Robert, Frédéric


Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^n$ ($n\geq 3$) such that $0\in\partial \Omega$. In this talk, we consider issues of non-existence, existence, and multiplicity of variational solutions for the borderline Dirichlet problem, $$ \left\{ \begin{array}{llll} -\Delta u-\gamma \frac{u}{|x|^2}- h(x) u &=& \frac{|u|^{2^\star(s)-2}u}{|x|^s} \ \ &\text{in } \Omega,\\ \hfill u&=&0 &\text{on }\ \partial\Omega, \end{array} \right.\eqno{(E)} $$ where $0<s<2$, $2^\star(s):=\frac{2(n-s)}{n-2}$, $\gamma\in\mathbb{R}$ and $h\in C^0(\overline{\Omega})$. We use sharp blow-up analysis on --possibly high energy-- solutions of corresponding subcritical problems to establish, for example, that if $\gamma<\frac{n^2}{4}-1$ and the principal curvatures of $\partial\Omega$ at $0$ are non-positive but not all of them vanishing, then Equation (E) has an infinite number of (possibly sign-changing) solutions. This complements results of the first and third authors, who showed in that if $\gamma\leq \frac{n^2}{4}-\frac{1}{4}$ and the mean curvature of $\partial\Omega$ at $0$ is negative, then (E) has a positive solution. On the other hand, our blow-up analysis also allows us to prove that if the mean curvature at $0$ is positive, then there is a surprising stability of regimes where there are no variational positive solutions under $C^1$-perturbations of the potential $h$. In particular, we show non-existence of such solutions for (E) whenever $\Omega$ is star-shaped and $h$ is close to $0$, which include situations not covered by the classical Pohozaev obstruction.

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