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Stable solutions to fractional semilinear equations: uniqueness, classification, and approximation results

Tomás Sanz-Perela

TL;DR

The paper studies stable solutions of the fractional semilinear equation $(-\Delta)^s u=f(u)$ on bounded domains with Dirichlet exterior data, focusing on convex nonlinearities $f$ and employing three solution notions: $L^1$-weak, energy in $H^s_\Omega$, and pointwise. It proves a uniqueness result for stable solutions, a classification in the zero-exterior, nonnegative case with $f(0)=0$, and an approximation framework that replaces $f$ by globally Lipschitz convex approximations $f_k$ to obtain bounded stable solutions $u_k$ converging to a given stable $u$ in $H^s_\Omega$; a secondary approximation via $(1-\varepsilon)f$ is also developed. The authors then transfer universal a priori estimates from the half-Laplacian to deduce interior regularity in low dimensions ($s=1/2$, $1\le n\le 4$). Finally, a counterexample shows that the energy-space membership is necessary for the approximation results. These results collectively provide tools to analyze stability-driven regularity and rigidity in nonlocal semilinear problems.

Abstract

We study stable solutions to fractional semilinear equations $(-Δ)^s u = f(u)$ in $Ω\subset \mathbb{R}^n$, for convex nonlinearities $f$, and under the Dirichlet exterior condition $u=g$ in $\mathbb{R}^n \setminus Ω$ with general $g$. We establish a uniqueness and a classification result, and we show that weak (energy) stable solutions can be approximated by a sequence of bounded (and hence regular) stable solutions to similar problems. As an application of our results, we establish the interior regularity of weak (energy) stable solutions to the problem for the half-Laplacian in dimensions $1 \leq n \leq 4$.

Stable solutions to fractional semilinear equations: uniqueness, classification, and approximation results

TL;DR

The paper studies stable solutions of the fractional semilinear equation on bounded domains with Dirichlet exterior data, focusing on convex nonlinearities and employing three solution notions: -weak, energy in , and pointwise. It proves a uniqueness result for stable solutions, a classification in the zero-exterior, nonnegative case with , and an approximation framework that replaces by globally Lipschitz convex approximations to obtain bounded stable solutions converging to a given stable in ; a secondary approximation via is also developed. The authors then transfer universal a priori estimates from the half-Laplacian to deduce interior regularity in low dimensions (, ). Finally, a counterexample shows that the energy-space membership is necessary for the approximation results. These results collectively provide tools to analyze stability-driven regularity and rigidity in nonlocal semilinear problems.

Abstract

We study stable solutions to fractional semilinear equations in , for convex nonlinearities , and under the Dirichlet exterior condition in with general . We establish a uniqueness and a classification result, and we show that weak (energy) stable solutions can be approximated by a sequence of bounded (and hence regular) stable solutions to similar problems. As an application of our results, we establish the interior regularity of weak (energy) stable solutions to the problem for the half-Laplacian in dimensions .
Paper Structure (12 sections, 18 theorems, 132 equations)

This paper contains 12 sections, 18 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Definitions and notation
  3. Main results
  4. Organization of the paper:
  5. Uniqueness of stable solutions
  6. A classification result
  7. Approximation of stable solutions
  8. Application: regularity of stable solutions in low dimensions
  9. Unbounded $L^1$-weak stable solutions in low dimensions
  10. An "intermediate value theorem" in fractional Sobolev spaces
  11. Some comments on the exterior condition in the notions of solution
  12. $L^1$-theory for the fractional Laplacian

Key Result

Theorem 1.7

Let $\Omega\subset \mathbb{R}^n$ a bounded smooth domain of $\mathbb{R}^n$ and let $f\in C^1(\mathbb{R})$ be a strictly convex function. Then, there exists at most one stable solution $u$ to Eq:SemilinearProblem ---in the $L^1$-weak sense and for $g$ as in Eq:Assumptionsg--- such that $u$ is bounded

Theorems & Definitions (48)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Definition 1.4
  • Remark 1.5
  • Definition 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Corollary 1.10
  • ...and 38 more