Non-existence of dead cores in fully nonlinear elliptic models

We investigate non-existence of nonnegative dead-core solutions for the problem $$\mathrm{\mid }Du{\mathrm{\mid }}^{\gamma }F(x,D^{2}u)+a(x)u^q=0in\mathrm{\Omega },u=0on\mathrm{\partial }\mathrm{\Omega }\mathrm{.}$$ Here $\Omega \subset $\mathbb{R}$^N$isaboundedsmoothdomain,$F$ is a fully nonlinear elliptic operator, $a:\mathrm{\Omega }\to \mathbb{R}$ 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} $\mathrm{T}\mathrm{r}$($\mathrm{A}$(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.