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Uniqueness and multiplicity for semilinear elliptic problems in unbounded domains

Henri Berestycki, Cole Graham, Juncheng Wei

TL;DR

This work analyzes positive bounded solutions of the semilinear elliptic equation $-\Delta u=f(u)$ in unbounded domains with Dirichlet boundary conditions, focusing on bistable and field-type nonlinearities. Geometry of the domain crucially governs solution multiplicity: epigraphs bounded from below admit a unique positive bounded solution (monotone in $y$), whereas epigraphs containing a cone of aperture larger than $\pi$ admit infinitely many, constructed via a Delaunay-ground-state framework and a Lyapunov–Schmidt reduction that balances boundary repulsion and inter-spike attraction. A key finding is that, despite potential multiplicity, every epigraph has at most one strictly stable solution; stability is characterized by the generalized principal eigenvalue and the asymptotic one-dimensional limits. The paper combines a refined moving planes technique with a stable-compact decomposition to prove uniqueness, and leverages connections to constant-mean-curvature surface theory to create many solutions in wide-cone domains; spike configurations are shown to be unstable, underscoring the special role of domain geometry in the qualitative behavior of nonlinear elliptic PDEs.

Abstract

We study the influence of geometry on semilinear elliptic equations of bistable or nonlinear-field type in unbounded domains. We discover a surprising dichotomy between epigraphs that are bounded from below and those that contain a cone of aperture greater than $π$: the former admit at most one positive bounded solution, while the latter support infinitely many. Nonetheless, we show that every epigraph admits at most one strictly stable solution. To prove uniqueness, we strengthen the method of moving planes by decomposing the domain into one region where solutions are stable and another where they enjoy a form of compactness. Our construction of many solutions exploits a connection with Delaunay surfaces in differential geometry, and extends to all domains containing a suitably wide cone, including exterior domains.

Uniqueness and multiplicity for semilinear elliptic problems in unbounded domains

TL;DR

This work analyzes positive bounded solutions of the semilinear elliptic equation in unbounded domains with Dirichlet boundary conditions, focusing on bistable and field-type nonlinearities. Geometry of the domain crucially governs solution multiplicity: epigraphs bounded from below admit a unique positive bounded solution (monotone in ), whereas epigraphs containing a cone of aperture larger than admit infinitely many, constructed via a Delaunay-ground-state framework and a Lyapunov–Schmidt reduction that balances boundary repulsion and inter-spike attraction. A key finding is that, despite potential multiplicity, every epigraph has at most one strictly stable solution; stability is characterized by the generalized principal eigenvalue and the asymptotic one-dimensional limits. The paper combines a refined moving planes technique with a stable-compact decomposition to prove uniqueness, and leverages connections to constant-mean-curvature surface theory to create many solutions in wide-cone domains; spike configurations are shown to be unstable, underscoring the special role of domain geometry in the qualitative behavior of nonlinear elliptic PDEs.

Abstract

We study the influence of geometry on semilinear elliptic equations of bistable or nonlinear-field type in unbounded domains. We discover a surprising dichotomy between epigraphs that are bounded from below and those that contain a cone of aperture greater than : the former admit at most one positive bounded solution, while the latter support infinitely many. Nonetheless, we show that every epigraph admits at most one strictly stable solution. To prove uniqueness, we strengthen the method of moving planes by decomposing the domain into one region where solutions are stable and another where they enjoy a form of compactness. Our construction of many solutions exploits a connection with Delaunay surfaces in differential geometry, and extends to all domains containing a suitably wide cone, including exterior domains.

Paper Structure

This paper contains 13 sections, 30 theorems, 107 equations, 2 figures.

Table of Contents

  1. Overview
  2. Uniqueness
  3. Multiplicity
  4. Preliminaries
  5. Delaunay solutions
  6. Approximate solution
  7. Controlling the projection
  8. Error and corrector
  9. Force balance
  10. Stability
  11. Stable solutions in epigraphs
  12. Instability of spike solutions
  13. Nondegeneracy

Key Result

Theorem 1.1

Let $f$ be a bistable nonlinearity satisfying hyp:bistable below. If $\Omega$ is a uniformly Lipschitz epigraph that is bounded from below, then eq:main admits a unique positive bounded solution $u$. It satisfies $\partial_y u > 0$ and $u(x) \to 1$ uniformly as $\operatorname{dist}(x, \Omega^\mathrm

Figures (2)

  • Figure 1: An equilibrium configuration of ground states (green) in a cone $\Omega$ of aperture greater than $\pi$. Arrows indicate boundary repulsion (red) and inter-state attraction (blue).
  • Figure 2: (a) Bistable and (b) field-type nonlinearities.

Theorems & Definitions (49)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • proof
  • Proposition 2.3
  • ...and 39 more