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A blow-up approach for a priori bounds in semilinear planar elliptic systems: the Brezis-Merle critical case

Laura Baldelli, Gabriele Mancini, Giulio Romani

TL;DR

We address the problem of obtaining uniform $L^{\infty}$-bounds for the planar semilinear Hamiltonian system $-\Delta u=f(v)$, $-\Delta v=g(u)$ under Dirichlet boundary conditions in a ball, in the Brezis–Merle critical regime. The main method is a novel blow-up analysis in which a carefully chosen scaling tied by a compatibility condition yields a Liouville-type limit in $\mathbb{R}^2$, complemented by integral estimates and Liouville classifications to rule out blow-up. This work extends the scalar Brezis–Merle theory to Hamiltonian two-component systems and uses Fixed Point Index theory to deduce the existence of a positive solution once a priori bounds are secured. It provides a comprehensive framework for critical exponential growth in 2D, with open questions on sharpness, domain generality, and possible mass-quantization phenomena.

Abstract

We establish uniform a priori estimates for solutions of semilinear planar Hamiltonian elliptic systems in a ball with Dirichlet boundary conditions. We consider a broad class of coupled nonlinearities with asymptotic critical behaviour in the sense of Brezis--Merle. The approach we follow is based on a blow-up analysis combined with Liouville--type theorems and integral estimates. Our results extend the scalar theory of uniform a priori bounds to the Hamiltonian case, and solve an open problem in [de Figueiredo D.G., do Ó J.M., Ruf B., Adv. Nonlinear Stud. 6 (2006), no. 2]. We believe that this approach is new in this setting. As a consequence of our a priori estimates, we prove the existence of a positive solution by means of Fixed Point Index theory.

A blow-up approach for a priori bounds in semilinear planar elliptic systems: the Brezis-Merle critical case

TL;DR

We address the problem of obtaining uniform -bounds for the planar semilinear Hamiltonian system , under Dirichlet boundary conditions in a ball, in the Brezis–Merle critical regime. The main method is a novel blow-up analysis in which a carefully chosen scaling tied by a compatibility condition yields a Liouville-type limit in , complemented by integral estimates and Liouville classifications to rule out blow-up. This work extends the scalar Brezis–Merle theory to Hamiltonian two-component systems and uses Fixed Point Index theory to deduce the existence of a positive solution once a priori bounds are secured. It provides a comprehensive framework for critical exponential growth in 2D, with open questions on sharpness, domain generality, and possible mass-quantization phenomena.

Abstract

We establish uniform a priori estimates for solutions of semilinear planar Hamiltonian elliptic systems in a ball with Dirichlet boundary conditions. We consider a broad class of coupled nonlinearities with asymptotic critical behaviour in the sense of Brezis--Merle. The approach we follow is based on a blow-up analysis combined with Liouville--type theorems and integral estimates. Our results extend the scalar theory of uniform a priori bounds to the Hamiltonian case, and solve an open problem in [de Figueiredo D.G., do Ó J.M., Ruf B., Adv. Nonlinear Stud. 6 (2006), no. 2]. We believe that this approach is new in this setting. As a consequence of our a priori estimates, we prove the existence of a positive solution by means of Fixed Point Index theory.
Paper Structure (17 sections, 14 theorems, 176 equations, 2 figures)

This paper contains 17 sections, 14 theorems, 176 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Assumptions and main results
  3. Notation.
  4. Preliminary results
  5. Elliptic regularity estimates
  6. Consequences of the assumptions
  7. Scaling and compatibility condition
  8. The scaling
  9. The compatibility condition
  10. The blow-up argument: Proof of Theorem \ref{['Thm_Uniform_bound']}
  11. Case 1
  12. Case 2
  13. Case 3
  14. Subcase 1
  15. Subcase 2
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $\Omega=B_1(0)$ be the unit ball in $\mathbb R^2$, assume conditions $(H_1)$, $(H_2)$, $(H_3)$, $(H_b)$, and alternatively Then there exists a constant $C>0$ such that for all (eventual) solutions $(u,v)$ of system sys.

Figures (2)

  • Figure 1: The state of the art on uniform a priori bounds for \ref{['sys']} for the model case \ref{['NL_model']}.
  • Figure 2: A qualitative sketch of the implicit function $t=t(s)$ in blue colour; the arrows indicate the growth of the function $J$ from negative to positive.

Theorems & Definitions (27)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1: Serrin, Theorem 2
  • Lemma 2.2: Serrin, Theorem 6
  • Proposition 2.3: Theorem 1.2, dFdOR
  • Remark 1
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 17 more