Uniform estimates for positive solutions of semilinear elliptic equations and related Liouville and one-dimensional symmetry results

TL;DR

This paper establishes uniform pointwise estimates for positive solutions of semilinear elliptic equations, removing previous restrictions, and applies these results to Liouville theorems, symmetry properties, and boundary behavior in various domains.

Contribution

It introduces a refined approach that removes restrictive assumptions and generalizes classical results for semilinear elliptic equations, including Liouville theorems and symmetry properties.

Findings

Uniform estimates for solutions independent of domain size

New proofs of Liouville theorems without boundary blow-up solutions

Confirmation of Gibbons' conjecture in two dimensions

Abstract

We consider a semilinear elliptic equation with Dirichlet boundary conditions in a smooth, possibly unbounded, domain. Under suitable assumptions, we deduce a condition on the size of the domain that implies the existence of a positive solution satisfying a uniform pointwise estimate. Here, uniform means that the estimate is independent of the domain. The main advantage of our approach is that it allows us to remove a restrictive monotonicity assumption that was imposed in the recent paper. In addition, we can remove a non-degeneracy condition that was assumed in the latter reference. Furthermore, we can generalize an old result, concerning semilinear elliptic nonlinear eigenvalue problems. Moreover, we study the boundary layer of global minimizers of the corresponding singular perturbation problem. For the above applications, our approach is based on a refinement of a result, concerning the behavior of global minimizers of the associated energy over large balls, subject to Dirichlet conditions. Combining this refinement with global bifurcation theory and the sliding method, we can prove uniform estimates for solutions away from their nodal set. Combining our approach with a-priori estimates that we obtain by blow-up, a doubling lemma, and known Liouville type theorems, we can give a new proof of a known Liouville type theorem without using boundary blow-up solutions. We can also provide an alternative proof, and a useful extension, of a Liouville theorem, involving the presence of an obstacle. Making use of the latter extension, we consider the singular perturbation problem with mixed boundary conditions. Moreover, we prove some new one-dimensional symmetry and rigidity properties of certain entire solutions to Allen-Cahn type equations, as well as in half spaces, convex cylindrical domains. In particular, we provide a new proof of Gibbons' conjecture in two dimensions.