Non-existence of dead cores in fully nonlinear elliptic models

Abstract

We investigate non-existence of nonnegative dead-core solutions for the problem $$|Du|^\gamma F(x, D^2u)+a(x)u^q = 0 \quad \mbox{in} \quad \Omega, \quad u=0 \quad \mbox{ on } \quad \partial\Omega.$$ 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, $\gamma \geq 0$, 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 $|D u(x)|^{\gamma} \mathrm{Tr}(\mathrm{A}(x) D^2 u(x)) + a(x)u^{q}(x) = 0$, where $\mathrm{A} \in C^0(\Omega;Sym(N))$ is a uniformly elliptic and non-negative matrix.