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Concentration Estimates for Emden-Fowler Equations with Boundary Singularities and Critical Growth

Nassif Ghoussoub, Frederic Robert

TL;DR

The paper analyzes Emden–Fowler equations with boundary singularities and critical Hardy–Sobolev growth on smooth bounded domains with 0 on the boundary. It develops a refined blow-up/concentration framework, reducing boundary concentration to half-space limits and employing Pohozaev identities to obtain compactness under favorable boundary geometry. It proves that the Hardy–Sobolev best constant is attained when the boundary mean curvature at 0 is negative and that there are infinitely many solutions under a local concavity condition on ∂Ω. The results extend Yamabe-type theory to boundary-singular critical problems and provide robust concentration-compactness tools for boundary phenomena, including detailed Appendix A–C on regularity, Green’s functions, and symmetry of limit profiles.

Abstract

We establish -among other things- existence and multiplicity of solutions for the Dirichlet problem $\sum_i\partial_{ii}u+\frac{|u|^{\crit-2}u}{|x|^s}=0$ on smooth bounded domains $\Omega$ of $ \rn$ ($n\geq 3$) involving the critical Hardy-Sobolev exponent $\crit =\frac{2(n-s)}{n-2}$ where $0<s<2$, and in the case where zero (the point of singularity) is on the boundary $\partial \Omega$. Just as in the Yamabe-type non-singular framework (i.e., when s=0), there is no nontrivial solution under global convexity assumption (e.g., when $\Omega$ is star-shaped around 0). However, in contrast to the non-satisfactory situation of the non-singular case, we show the existence of an infinite number of solutions under an assumption of local strict concavity of $\partial \Omega$ at 0 in at least one direction. More precisely, we need the principal curvatures of $\partial \Omega$ at 0 to be non-positive but not all vanishing. We also show that the best constant in the Hardy-Sobolev inequality is attained as long as the mean curvature of $\partial \Omega$ at 0 is negative, extending the results of [21] and completing our result of [22] to include dimension 3. The key ingredients in our proof are refined concentration estimates which yield compactness for certain Palais-Smale sequences which do not hold in the non-singular case.

Concentration Estimates for Emden-Fowler Equations with Boundary Singularities and Critical Growth

TL;DR

The paper analyzes Emden–Fowler equations with boundary singularities and critical Hardy–Sobolev growth on smooth bounded domains with 0 on the boundary. It develops a refined blow-up/concentration framework, reducing boundary concentration to half-space limits and employing Pohozaev identities to obtain compactness under favorable boundary geometry. It proves that the Hardy–Sobolev best constant is attained when the boundary mean curvature at 0 is negative and that there are infinitely many solutions under a local concavity condition on ∂Ω. The results extend Yamabe-type theory to boundary-singular critical problems and provide robust concentration-compactness tools for boundary phenomena, including detailed Appendix A–C on regularity, Green’s functions, and symmetry of limit profiles.

Abstract

We establish -among other things- existence and multiplicity of solutions for the Dirichlet problem on smooth bounded domains of () involving the critical Hardy-Sobolev exponent where , and in the case where zero (the point of singularity) is on the boundary . Just as in the Yamabe-type non-singular framework (i.e., when s=0), there is no nontrivial solution under global convexity assumption (e.g., when is star-shaped around 0). However, in contrast to the non-satisfactory situation of the non-singular case, we show the existence of an infinite number of solutions under an assumption of local strict concavity of at 0 in at least one direction. More precisely, we need the principal curvatures of at 0 to be non-positive but not all vanishing. We also show that the best constant in the Hardy-Sobolev inequality is attained as long as the mean curvature of at 0 is negative, extending the results of [21] and completing our result of [22] to include dimension 3. The key ingredients in our proof are refined concentration estimates which yield compactness for certain Palais-Smale sequences which do not hold in the non-singular case.

Paper Structure

This paper contains 12 sections, 21 theorems, 464 equations.

Table of Contents

  1. Introduction and statement of the results
  2. Basic facts and preliminary Lemmae
  3. Construction and exhaustion of the blow-up scales
  4. Strong pointwise estimates, Part 1
  5. Strong pointwise estimates, Part 2
  6. Pohozaev identity and proof of compactness
  7. Proof of existence and multiplicity
  8. Proof of Theorem 1.1
  9. Proof of Theorem 1.2
  10. Appendix A: Regularity of weak solutions
  11. Appendix B: Properties of the Green's function
  12. Appendix C: Symmetry of the positive solutions to the limit equation

Key Result

Theorem 1.1

Let \Omega be a smooth bounded oriented domain of \mathbb{R}^n, n\geq 3, such that 0\in\partial\Omega and assume s\in (0,2). If the mean curvature of \partial\Omega at 0 is negative, then \mu_{s}(\Omega) is achieved by a positive function which is --a positive multiple of-- a solution for

Theorems & Definitions (25)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1
  • Lemma 2.1
  • Lemma 2.2
  • Proposition 3.1
  • Lemma 3.1
  • Proposition 3.2
  • Proposition 3.3
  • ...and 15 more