Calculus of Variations and Geometric Measure Theory

N. Soave - T. Weth

The unique continuation property of sublinear equations

created by soave on 26 Jul 2017
modified on 01 Feb 2019


Published Paper

Inserted: 26 jul 2017
Last Updated: 1 feb 2019

Journal: SIAM Journal of Mathematical Analysis
Volume: 50
Number: 4
Pages: 3919--3938
Year: 2018
Links: Link arXiv


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 $-\Delta u=
^{\sigma−1}u$ 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.