  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.
