Uniqueness and multiplicity for semilinear elliptic problems in unbounded domains

TL;DR

This paper investigates how the geometry of unbounded domains affects the number and stability of solutions to semilinear elliptic equations, revealing a geometric dichotomy in solution multiplicity.

Contribution

It establishes a surprising geometric dichotomy in solution multiplicity for semilinear elliptic problems based on domain shape, and extends the analysis using geometric and stability methods.

Findings

Bounded-from-below epigraphs admit at most one positive bounded solution.

Epigraphs containing a cone with aperture greater than π support infinitely many solutions.

Every epigraph admits at most one strictly stable solution.

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 $\pi$: 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.