The Hardy–Schrödinger operator on the Poincaré ball: Compactness, multiplicity, and stability of the Pohozaev obstruction
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Cited by 4 publications
References 21 publications
Heat and Martin Kernel estimates for Schrödinger operators with critical Hardy potentials
Let $$\Omega $$ Ω be a bounded domain in $${{\mathbb {R}}}^N$$ R N with $$C^2$$ C 2 boundary and let $$K\subset \partial \Omega $$ K ⊂ ∂ Ω be either a $$C^2$$ C 2 submanifold of the boundary of codimension $$k<N$$ k < N or a point. In this article we study various problems related to the Schrödinger operator $$L_{\mu } =-\Delta - \mu d_K^{-2}$$ L μ = - Δ - μ d K - 2 where $$d_K$$ d K denotes the distance to K and $$\mu \le k^2/4$$ μ ≤ k 2 / 4 . We establish parabolic boundary Harnack inequalities as well as related two-sided heat kernel and Green function estimates. We construct the associated Martin kernel and prove existence and uniqueness for the corresponding boundary value problem with data given by measures. To prove our results we introduce among other things a suitable notion of boundary trace. This trace is different from the one used by Marcus and Nguyen (Math Ann 374(1–2):361–394, 2019) thus allowing us to cover the whole range $$\mu \le k^2/4$$ μ ≤ k 2 / 4 .
Heat and Martin Kernel estimates for Schrödinger operators with critical Hardy potentials
Let $$\Omega $$ Ω be a bounded domain in $${{\mathbb {R}}}^N$$ R N with $$C^2$$ C 2 boundary and let $$K\subset \partial \Omega $$ K ⊂ ∂ Ω be either a $$C^2$$ C 2 submanifold of the boundary of codimension $$k<N$$ k < N or a point. In this article we study various problems related to the Schrödinger operator $$L_{\mu } =-\Delta - \mu d_K^{-2}$$ L μ = - Δ - μ d K - 2 where $$d_K$$ d K denotes the distance to K and $$\mu \le k^2/4$$ μ ≤ k 2 / 4 . We establish parabolic boundary Harnack inequalities as well as related two-sided heat kernel and Green function estimates. We construct the associated Martin kernel and prove existence and uniqueness for the corresponding boundary value problem with data given by measures. To prove our results we introduce among other things a suitable notion of boundary trace. This trace is different from the one used by Marcus and Nguyen (Math Ann 374(1–2):361–394, 2019) thus allowing us to cover the whole range $$\mu \le k^2/4$$ μ ≤ k 2 / 4 .
Multiplicity and Stability of the Pohozaev Obstruction for Hardy-Schrödinger Equations with Boundary Singularity
Let Ω \Omega be a smooth bounded domain in R n \mathbb {R}^n ( n ≥ 3 n\geq 3 ) such that 0 ∈ ∂ Ω 0\in \partial \Omega . We consider issues of non-existence, existence, and multiplicity of variational solutions in H 1 , 0 2 ( Ω ) H_{1,0}^2(\Omega ) for the borderline Dirichlet problem, { − Δ u − γ u | x | 2 − h ( x ) u a m p ; = a m p ; | u | 2 ⋆ ( s ) − 2 u | x | s a m p ; in Ω , u a m p ; = a m p ; 0 a m p ; on ∂ Ω ∖ { 0 } , \begin{equation*} \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 \setminus \{ 0 \} , \end{array} \right . \end{equation*} where 0 > s > 2 0>s>2 , 2 ⋆ ( s ) ≔ 2 ( n − s ) n − 2 {2^\star (s)}≔\frac {2(n-s)}{n-2} , γ ∈ R \gamma \in \mathbb {R} and h ∈ C 0 ( Ω ¯ ) 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 γ > n 2 4 − 1 \gamma >\frac {n^2}{4}-1 and the principal curvatures of ∂ Ω \partial \Omega at 0 0 are non-positive but not all of them vanishing, then Equation (E) has an infinite number of high energy (possibly sign-changing) solutions in H 1 , 0 2 ( Ω ) H_{1,0}^2(\Omega ) . This complements results of the first and third authors, who showed in their 2016 article, Hardy-Singular Boundary Mass and Sobolev-Critical Variational Problems, that if γ ≤ n 2 4 − 1 4 \gamma \leq \frac {n^2}{4}-\frac {1}{4} and the mean curvature of ∂ Ω \partial \Omega at 0 0 is negative, then ( E ) (E) has a positive least energy solution. On the other hand, the sharp blow-up analysis also allows us to show that if the mean curvature at 0 0 is nonzero and the mass, when defined, is also nonzero, then there is a surprising stability of regimes where there are no variational positive solutions under C 1 C^1 -perturbations of the potential h h . In particular, and in sharp contrast with the non-singular case (i.e., when γ = s = 0 \gamma =s=0 ), we prove non-existence of such solutions for ( E ) (E) in any dimension, whenever Ω \Omega is star-shaped and h h is close to 0 0 , which include situations not covered by the classical Pohozaev obstruction.
Extremal functions for twisted sharp Sobolev inequalities with lower order remainder terms. The high-dimensional case
We prove existence of extremal functions and compactness of the set of extremal functions for twisted sharp 𝑛-dimensional Sobolev inequalities with lower order remainder terms when n ≥ 5 n\geq 5 .
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