Non-existence of dead cores in fully nonlinear elliptic models

Abstract

We investigate non-existence of nonnegative dead-core solutions for the problem $$|Du|^γF(x, D^2u)+a(x)u^q = 0 \quad \mbox{in} \quad Ω, \quad u=0 \quad \mbox{ on } \quad \partialΩ.$$ Here $\Omega \subset {\mathbb{R}}^N$ is a bounded smooth domain, $F$ is a fully nonlinear elliptic operator, $a:\Omega \to \mathbb{R}$ is a sign-changing weight, $\gamma \ge 0$, and $0<q<\gamma +1$. We show that this problem has no non-trivial dead core solutions if either $q$ is close enough to $\gamma +1$ or the negative part of $a$ is sufficiently small. In addition, we obtain the existence and uniqueness of a positive solution under these conditions on $q$ and $a$. Our results extend previous ones established in the semilinear case, and are new even for the simple model $|Du(x){|}^{\gamma }\mathrm{Tr}(\mathrm{A}(x)D^{2}u(x))+a(x)u^q(x)=0$, where $\mathrm{A}\in C^0(\Omega ;Sym(N))$ is a uniformly elliptic and non-negative matrix.