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Article Dans Une Revue Calculus of Variations and Partial Differential Equations Année : 2017

The Hardy-Schrödinger operator with interior singularity: The remaining cases

Résumé

We consider the remaining unsettled cases in the problem of existence of energy minimizing solutions for the Dirichlet value problem $L_\gamma u-\lambda u=\frac{u^{2^*(s)-1}}{|x|^s}$ on a smooth bounded domain $\Omega$ in $\mathbb{R}^n$ ($n\geq 3$) having the singularity $0$ in its interior. Here $\gamma <\frac{(n-2)^2}{4}$, $0\leq s <2$, $2^*(s):=\frac{2(n-s)}{n-2}$ and $0\leq \lambda <\lambda_1(L_\gamma)$, the latter being the first eigenvalue of the Hardy-Schr\"odinger operator $L_\gamma:=-\Delta -\frac{\gamma}{|x|^2}$. There is a threshold $\lambda^*(\gamma, \Omega) \geq 0$ beyond which the minimal energy is achieved, but below which, it is not. It is well known that $\lambda^*(\Omega) = 0$ in higher dimensions, for example if $0\leq \gamma \leq \frac{(n-2)^2}{4}-1$. Our main objective in this paper is to show that this threshold is strictly positive in ``lower dimensions" such as when $ \frac{(n-2)^2}{4}-1<\gamma <\frac{(n-2)^2}{4}$, to identify the critical dimensions (i.e., when the situation changes), and to characterize it in terms of $\Omega$ and $\gamma$. If either $s>0$ or if $\gamma > 0$, i.e., in {\it the truly singular case}, we show that in low dimensions, a solution is guaranteed by the positivity of the ``Hardy-singular internal mass" of $\Omega$, a notion that we introduce herein. On the other hand, and just like the case wnen $\gamma=s=0$ studied by Brezis-Nirenberg and completed by Druet, $n=3$ is the critical dimension, and the classical positive mass theorem is sufficient for the {\it merely singular case}, that is when $s=0$, $\gamma \leq 0$.
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Dates et versions

hal-01422509 , version 1 (26-12-2016)
hal-01422509 , version 2 (18-09-2017)

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Nassif Ghoussoub, Frédéric Robert. The Hardy-Schrödinger operator with interior singularity: The remaining cases. Calculus of Variations and Partial Differential Equations, 2017, 56, ⟨10.1007/s00526-017-1238-1⟩. ⟨hal-01422509v2⟩
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