Calculus of Variations and Geometric Measure Theory
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N. Soave - T. Weth

The unique continuation property of sublinear equations

created by soave on 26 Jul 2017

[BibTeX]

Submitted Paper

Inserted: 26 jul 2017
Last Updated: 26 jul 2017

Year: 2017
Links: Link arXiv

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.

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