A computer-assisted existence proof for Emden's equation on an unbounded L-shaped domain

TL;DR

This paper demonstrates the existence, uniqueness, and decay properties of a solution to Emden's equation on an unbounded L-shaped domain using computer-assisted proofs, serving as a foundation for future solutions on expanding domains.

Contribution

It introduces a computer-assisted fixed-point method to rigorously prove the existence and non-degeneracy of solutions to Emden's equation on complex unbounded domains.

Findings

Existence of a non-trivial solution on an unbounded L-shaped domain.

Proof of exponential decay at infinity of the solution.

Establishment of non-degeneracy through eigenvalue bounds.

Abstract

We prove existence, non-degeneracy, and exponential decay at infinity of a non-trivial solution to Emden's equation $-\Delta u = | u |^3$ on an unbounded $L$-shaped domain, subject to Dirichlet boundary conditions. Besides the direct value of this result, we also regard this solution as a building block for solutions on expanding bounded domains with corners, to be established in future work. Our proof makes heavy use of computer assistance: Starting from a numerical approximate solution, we use a fixed-point argument to prove existence of a near-by exact solution. The eigenvalue bounds established in the course of this proof also imply non-degeneracy of the solution.