A Computer-Assisted Uniqueness Proof for a Semilinear Elliptic Boundary Value Problem

TL;DR

This paper presents a computer-assisted proof of the uniqueness of positive solutions for a semilinear elliptic boundary value problem with p=3, extending previous results and reducing computational complexity.

Contribution

The authors develop a more efficient computational method to prove uniqueness for p=3, advancing the application of computer-assisted proofs in nonlinear elliptic PDEs.

Findings

Proved uniqueness for p=3 using computer-assisted methods.

Reduced computational complexity compared to previous approaches.

Extended the scope of computer-assisted proofs in elliptic boundary value problems.

Abstract

A wide variety of articles, starting with the famous paper (Gidas, Ni and Nirenberg in Commun. Math. Phys. 68, 209-243 (1979)) is devoted to the uniqueness question for the semilinear elliptic boundary value problem -{\Delta}u={\lambda}u+u^p in {\Omega}, u>0 in {\Omega}, u=0 on the boundary of {\Omega}, where {\lambda} ranges between 0 and the first Dirichlet Laplacian eigenvalue. So far, this question was settled in the case of {\Omega} being a ball and, for more general domains, in the case {\lambda}=0. In (McKenna et al. in J. Differ. Equ. 247, 2140-2162 (2009)), we proposed a computer-assisted approach to this uniqueness question, which indeed provided a proof in the case {\Omega}=(0,1)x(0,1), and p=2. Due to the high numerical complexity, we were not able in (McKenna et al. in J. Differ. Equ. 247, 2140-2162 (2009)) to treat higher values of p. Here, by a significant reduction of the complexity, we will prove uniqueness for the case p=3.