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Concentration analysis for elliptic critical equations with no boundary control: ground-state blow-up

Hussein Mesmar, Frédéric Robert

TL;DR

This work analyzes ground-state blow-up for critical elliptic equations on manifolds with boundary without boundary control, under $L^2$-concentration. The authors establish precise blow-up profiles: after rescaling, solutions converge to the standard bubble $w$ in $C^2_{\mathrm{loc}}(\mathbb{R}^n)$ with energy concentration $\int_M f u_\varepsilon^{2^*}\, dv_g \to K$-dependent constant, and show that the blow-up point $x_0$ is a critical point of $f$ with $d_g(x_\varepsilon,x_0)=o(\mu_\varepsilon)$; for $n\in\{4,5,6\}$ they relate $h(x_0)$ to scalar curvature and $\Delta_g f/f$. The paper further extends the analysis to $G$-invariant (symmetry) problems, yielding a reduction to a quotient manifold and a corresponding scalar-curvature relation in the reduced setting, thereby providing a framework for fast convergence of concentration points and applications to supercritical problems with symmetries.

Abstract

We perform the apriori analysis of solutions to critical nonlinear elliptic equations on manifolds with boundary. The solutions are of minimizing type. The originality is that we impose no condition on the boundary, which leads us to assume $L^2-$concentration. We also analyze the effect of a non-homogeneous nonlinearity that results in the fast convergence of the concentration point.

Concentration analysis for elliptic critical equations with no boundary control: ground-state blow-up

TL;DR

This work analyzes ground-state blow-up for critical elliptic equations on manifolds with boundary without boundary control, under -concentration. The authors establish precise blow-up profiles: after rescaling, solutions converge to the standard bubble in with energy concentration -dependent constant, and show that the blow-up point is a critical point of with ; for they relate to scalar curvature and . The paper further extends the analysis to -invariant (symmetry) problems, yielding a reduction to a quotient manifold and a corresponding scalar-curvature relation in the reduced setting, thereby providing a framework for fast convergence of concentration points and applications to supercritical problems with symmetries.

Abstract

We perform the apriori analysis of solutions to critical nonlinear elliptic equations on manifolds with boundary. The solutions are of minimizing type. The originality is that we impose no condition on the boundary, which leads us to assume concentration. We also analyze the effect of a non-homogeneous nonlinearity that results in the fast convergence of the concentration point.
Paper Structure (7 sections, 6 theorems, 151 equations)

This paper contains 7 sections, 6 theorems, 151 equations.

Table of Contents

  1. Introduction
  2. Context and main results
  3. Application to supercritical problems with symmetries
  4. Pointwise control
  5. Speed of convergence of $(x_\epsilon)_\epsilon$
  6. Interaction with the scalar curvature: proof of \ref{['relation:de:h0:avec:courbure:scalaire:cas:variete:a:bord']}
  7. Application to a super-critical problem: proof of Theorem \ref{['th:mesmar:2:supercitical:case:blow:up']}

Key Result

Theorem 1

Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n\geq 4$ with nonempty boundary $\partial M\neq\emptyset$. We fix $f\in C^2(\overline{M} )$ such that $f>0$. We consider a family $(h_\epsilon)_\epsilon\in C^1(\overline{M} )$ and $f \in C^2(\overline{M})$, $f>0$, such that there exis We let $(u_\epsilon)_\epsilon\in C^2(\overline{M} )$ be a family of solutions to Let $x_\epsilon\i

Theorems & Definitions (11)

  • Theorem 1
  • Theorem 2
  • Claim 1
  • Claim 2
  • Claim 3
  • Proposition 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Claim 4
  • ...and 1 more