The unique continuation property of sublinear equations

TL;DR

This paper proves the unique continuation property for a class of semi-linear elliptic equations with sublinear nonlinearities, showing solutions that vanish in an open set must vanish everywhere, extending to variable coefficients.

Contribution

It establishes the unique continuation property for sublinear elliptic equations, including variable coefficient cases, which was previously not well understood.

Findings

Solutions vanish everywhere if they vanish in an open subset.

Results apply to equations with variable coefficients and inhomogeneous terms.

The paper extends unique continuation results to sublinear nonlinearities.

Abstract

We derive the unique continuation property of a class of semi-linear elliptic equations with non-Lipschitz nonlinearities. The simplest type of equations to which our results apply is given as $-\Delta u = |u|^{\sigma-1} u$ in a domain $\Omega \subset \mathbb{R}^N$, with $0 \le \sigma <1$. Despite the sublinear character of the nonlinear term, we prove that if a solution vanishes in an open subset of $\Omega$, then it vanishes necessarily in the whole $\Omega$. We then extend the result to equations with variable coefficients operators and inhomogeneous right-hand side.