The Unique Continuation Property of Sublinear Equations
Nicola Soave, Tobias Weth · SIAM Journal on Mathematical Analysis · 2018
We derive the unique continuation property of a class of semilinear 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.