Uniqueness for linear integro-differential equations in the real line and applications

TL;DR

This paper establishes the uniqueness of solutions for certain nonlocal linear equations involving elliptic integro-differential operators on the real line, and applies these results to prove nondegeneracy of layer solutions in a nonlocal Allen-Cahn type problem.

Contribution

It provides the first nonlocal uniqueness and nondegeneracy results without relying on the Caffarelli-Silvestre extension technique.

Findings

Uniqueness of solutions for nonlocal linear equations in one dimension.

Nondegeneracy of layer solutions to nonlocal Allen-Cahn equations.

Application of a nonlocal Liouville-type method to prove these results.

Abstract

In this work we prove the uniqueness of solutions to the nonlocal linear equation $L \varphi - c(x)\varphi = 0$ in $\mathbb{R}$, where $L$ is an elliptic integro-differential operator, in the presence of a positive solution or of an odd solution vanishing only at zero. As an application, we deduce the nondegeneracy of layer solutions (bounded and monotone solutions) to the semilinear problem $L u = f(u)$ in $\mathbb{R}$ when the nonlinearity is of Allen-Cahn type. To our knowledge, this is the first work where such uniqueness and nondegeneracy results are proven in the nonlocal framework when the Caffarelli-Silvestre extension technique is not available. Our proofs are based on a nonlocal Liouville-type method developed by Hamel, Ros-Oton, Sire, and Valdinoci for nonlinear problems in dimension two.