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Monotonicity for solutions to semilinear problems in epigraphs

Nicolas Beuvin, Alberto Farina, Berardino Sciunzi

TL;DR

The paper proves that positive (possibly unbounded) solutions of the semilinear Poisson equation $-\Delta u=f(u)$ on lower-bounded epigraphs are strictly increasing in the vertical direction under broad geometric and regularity conditions. Central to the results are new comparison principles on unbounded domains with good directional sections and a tailored moving-plane method for epigraphs, complemented by uniform Hölder estimates in truncated regions. The authors extend monotonicity to merely continuous epigraphs via the class $\mathcal{G}$ and derive classification and nonexistence (Liouville-type) results in various dimensions, including sharp results when $f$ is convex/nondecreasing. The work unifies and extends previous half-space/coercive-epigraph results, providing robust tools for symmetry, uniqueness, and nonexistence in fairly general unbounded domains, with explicit applications to behavior at infinity and stability analyses.

Abstract

We consider positive solutions, possibly unbounded, to the semilinear equation $-Δu=f(u)$ on continuous epigraphs bounded from below. Under the homogeneous Dirichlet boundary condition, we prove new monotonicity results for $u$, when $f$ is a (locally or globally) Lipschitz-continuous function satisfying $ f(0) \geq 0$. As an application of our new monotonicity theorems, we prove some classification and/or non-existence results. To prove our results, we first establish some new comparison principles for semilinear problems on general unbounded open sets of $\mathbb{R}^N$, and then we use them to start and to complete a modified version of the moving plane method adapted to the geometry of the epigraph $Ω$. As a by-product of our analysis, we also prove some new results of uniqueness and symmetry for solutions (possibly unbounded and sign-changing) to the homogeneous Dirichlet BVP for the semilinear Poisson equation in fairly general unbounded domains.

Monotonicity for solutions to semilinear problems in epigraphs

TL;DR

The paper proves that positive (possibly unbounded) solutions of the semilinear Poisson equation on lower-bounded epigraphs are strictly increasing in the vertical direction under broad geometric and regularity conditions. Central to the results are new comparison principles on unbounded domains with good directional sections and a tailored moving-plane method for epigraphs, complemented by uniform Hölder estimates in truncated regions. The authors extend monotonicity to merely continuous epigraphs via the class and derive classification and nonexistence (Liouville-type) results in various dimensions, including sharp results when is convex/nondecreasing. The work unifies and extends previous half-space/coercive-epigraph results, providing robust tools for symmetry, uniqueness, and nonexistence in fairly general unbounded domains, with explicit applications to behavior at infinity and stability analyses.

Abstract

We consider positive solutions, possibly unbounded, to the semilinear equation on continuous epigraphs bounded from below. Under the homogeneous Dirichlet boundary condition, we prove new monotonicity results for , when is a (locally or globally) Lipschitz-continuous function satisfying . As an application of our new monotonicity theorems, we prove some classification and/or non-existence results. To prove our results, we first establish some new comparison principles for semilinear problems on general unbounded open sets of , and then we use them to start and to complete a modified version of the moving plane method adapted to the geometry of the epigraph . As a by-product of our analysis, we also prove some new results of uniqueness and symmetry for solutions (possibly unbounded and sign-changing) to the homogeneous Dirichlet BVP for the semilinear Poisson equation in fairly general unbounded domains.

Paper Structure

This paper contains 8 sections, 29 theorems, 187 equations, 14 figures.

Table of Contents

  1. Introduction and main results
  2. Some new comparison principles on unbounded domains and their applications to the uniqueness and symmetry of solutions to the semilinear Poisson equation
  3. Some uniform estimates in unbounded domains
  4. Proofs
  5. Extensions to merely continuous epigraphs bounded from below and further observations
  6. Some applications to classification and non-existence results
  7. Some examples
  8. Notations

Key Result

Theorem 1.1

Let $\Omega$ be a uniformly continuous epigraph bounded from below. Assume $f \in {Lip}_{loc}([0,+\infty))$ with and let $u \in C^0(\overline{\Omega}) \cap H^1_{loc}(\overline{\Omega})$ be a distributional solution to NonLin-PoissonEq which is uniformly continuous on finite strips.i.e., for any $R>0$, $u$ is uniformly continuous on the strip $\Omega \cap \left \lbrace x_N <R \right\rbrace$. Then

Figures (14)

  • Figure 1: $\Omega_{1}$
  • Figure 2: Case 2.1
  • Figure 3: Case 2.2
  • Figure 4: Case 2.3
  • Figure 5: Case 2.1
  • ...and 9 more figures

Theorems & Definitions (42)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 32 more