Holomorphic extension of solutions of semilinear elliptic equations

TL;DR

This paper demonstrates that solutions to a broad class of semilinear elliptic equations in real space can be extended to holomorphic functions in complex sectors, with exponential decay, enhancing previous results limited to strips.

Contribution

It extends the known holomorphic extension of solutions from strips to sectors for a wide class of semilinear elliptic equations, including pseudodifferential ones.

Findings

Solutions extend holomorphically to sectors in a8C^da9.

Extended solutions exhibit exponential decay.

Results apply to solitary wave solutions of classical nonlinear evolution equations.

Abstract

We prove, for a wide class of semilinear elliptic differential and pseudodifferential equations in $\R^d$, that the solutions which are sufficiently regular and have a certain decay at infinity extend to holomorphic functions in sectors of $\mathbb{C}^d$, improving earlier results where the extension was shown for a strip. Moreover, exponential decay for such extended solutions is also proved. The results apply, in particular, to solitary wave solutions of many classical nonlinear evolution equations.