Measure boundary value problems for semilinear elliptic equations with critical Hardy potentials

Gkikas, Konstantinos T.; Véron, Laurent

Comptes Rendus. Mathématique, Tome 353 (2015) no. 4, pp. 315-320.

Soient $\Omega \subset {\mathbb{R}}^N$ un domaine de classe $C^2$ et ${\mathcal{L}}_{\kappa }=-\mathrm{\Delta }-\frac{\kappa }{d^2}$ où $d=\mathrm{dist}(.,\partial \Omega )$ et $0<\kappa \le \frac{1}{4}$. Soient ${\alpha }_{\pm }=1\pm \sqrt{1-4\kappa }y,{\lambda }_{\kappa }$ la première valeur propre de ${\mathcal{L}}_{\kappa }$ et ${\varphi }_{\kappa }$ la fonction propre positive correspondante. Si g est une fonction continue croissante vérifiant ${\int }_1^{\infty }(g(s)+|g(-s)|)s^{-2\frac{2N-2+{\alpha }_{+}}{2N-4+{\alpha }_{+}}}ds<\infty$, alors pour toutes mesures de Radon $\nu \in {\mathfrak{M}}_{{\varphi }_{\kappa }}(\Omega )$ et $\mu \in \mathfrak{M}(\partial \Omega )$, il existe une unique solution faible au problème $P_{\nu ,\mu }$ : ${\mathcal{L}}_{\kappa }u+g(u)=\nu$ dans Ω, $u=\mu$ sur ∂Ω. Si $g(r)={|r|}^{q-1}u$ ($q>1$), nous démontrons qu'une condition nécessaire et suffisante pour résoudre $P_{0,\mu }$ avec $\mu >0$ est que μ soit absolument continue par rapport à la capacité associée à l'espace $B^{2-\frac{2+{\alpha }_{+}}{2q^{\prime }},q^{\prime }}({\mathbb{R}}^{N-1})$. Cette capacité caractérise les ensembles éliminables du bord. Dans le cas sous-critique, nous classifions les singularités isolées au bord des solutions positives.

Let $\Omega \subset {\mathbb{R}}^N$ be a bounded $C^2$ domain and ${\mathcal{L}}_{\kappa }=-\mathrm{\Delta }-\frac{\kappa }{d^2}$ where $d=\mathrm{dist}(.,\partial \Omega )$ and $0<\kappa \le \frac{1}{4}$. Let ${\alpha }_{\pm }=1\pm \sqrt{1-4\kappa }$, ${\lambda }_{\kappa }$ the first eigenvalue of ${\mathcal{L}}_{\kappa }$ with corresponding positive eigenfunction ${\varphi }_{\kappa }$. If g is a continuous nondecreasing function satisfying ${\int }_1^{\infty }(g(s)+|g(-s)|)s^{-2\frac{2N-2+{\alpha }_{+}}{2N-4+{\alpha }_{+}}}ds<\infty$, then for any Radon measures $\nu \in {\mathfrak{M}}_{{\varphi }_{\kappa }}(\Omega )$ and $\mu \in \mathfrak{M}(\partial \Omega )$ there exists a unique weak solution to problem $P_{\nu ,\mu }$: ${\mathcal{L}}_{\kappa }u+g(u)=\nu$ in Ω, $u=\mu$ on ∂Ω. If $g(r)={|r|}^{q-1}u$ ($q>1$), we prove that, in the supercritical range of q, a necessary and sufficient condition for solving $P_{0,\mu }$ with $\mu >0$ is that μ is absolutely continuous with respect to the capacity associated with the space $B^{2-\frac{2+{\alpha }_{+}}{2q^{\prime }},q^{\prime }}({\mathbb{R}}^{N-1})$. We also characterize the boundary removable sets in terms of this capacity. In the subcritical range of q we classify the isolated singularities of positive solutions.

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