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Rigidity for a semilinear Neumann problem with exponential nonlinearity in the large diffusion limit

Juneyoung Seo

Abstract

We consider a semilinear Neumann problem with exponential nonlinearity in a smooth bounded domain $Ω\subset \mathbb{R}^2$. We prove that there exists a threshold $\bar{\varepsilon}>0$ such that for all $\varepsilon>\bar{\varepsilon}$, any classical solution must be constant. This result provides a positive answer to a conjecture recently posed by Calanchi, Ciraolo, and Messina (2026). Our proof relies on a combination of $L^1$-estimates, a Jensen-type argument via the Neumann Green's function to obtain uniform exponential integrability, and elliptic regularity.

Rigidity for a semilinear Neumann problem with exponential nonlinearity in the large diffusion limit

Abstract

We consider a semilinear Neumann problem with exponential nonlinearity in a smooth bounded domain . We prove that there exists a threshold such that for all , any classical solution must be constant. This result provides a positive answer to a conjecture recently posed by Calanchi, Ciraolo, and Messina (2026). Our proof relies on a combination of -estimates, a Jensen-type argument via the Neumann Green's function to obtain uniform exponential integrability, and elliptic regularity.

Paper Structure

This paper contains 3 sections, 4 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Uniform A Priori Estimates
  3. Proof of the Main Theorem

Key Result

Theorem 1.1

Let $a > 1$. There exists a constant $\bar{\varepsilon} > 0$, depending only on $a$ and $\Omega$, such that for every $\varepsilon > \bar{\varepsilon}$, any classical solution $u$ to eq:main must be a constant. Consequently, the only solutions are $u \equiv 0$ and $u \equiv \xi_a$, where $\xi_a > 0$

Theorems & Definitions (8)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • proof