Hardy-Sobolev inequalities with singularities on non smooth boundary. Part 2: Influence of the global geometry in small dimensions
Abstract: We consider Hardy-Sobolev nonlinear equations on domains with singularities. We introduced this problem in Cheikh-Ali [4]. Under a local geometric hypothesis, namely that the generalized mean curvature is negative (see ( 7) below), we proved the existence of extremals for the relevant Hardy-Sobolev inequality for large dimensions. In the present paper, we tackle the question of small dimensions that was left open. We introduce a "mass", that is a global quantity, the positivity of which ensures the existence o… Show more
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Cited by 2 publications
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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.
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.
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