We investigate non-existence of non-negative dead core solutions for the problem $$|Du{|}^{\gamma }F(x,D^{2}u)+a(x)u^q=0\text{in }\mathrm{\Omega },u=0\text{on }\partial \mathrm{\Omega }.$$ Here, $\mathrm{\Omega }\subset {ℝ}^N$ is a bounded smooth domain, $F$ is a fully nonlinear elliptic operator, $a:\mathrm{\Omega }\to ℝ$ is a sign-changing weight, $\gamma \ge 0$, and $0<q<\gamma +1$. We show that this problem has no nontrivial 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 $|Du(x){|}^{\gamma }\mathrm{\text{Tr}}(\mathrm{\text{A}}(x)D^{2}u(x))+a(x)u^q(x)=0$, where $\mathrm{\text{A}}\in C^0(\mathrm{\Omega };\mathrm{\text{Sym}}(N))$ is a uniformly elliptic and non-negative matrix.

我们调查了该问题不存在非负死核解决方案$$|D你{|}^{\gamma }F(X,D^2你)+\mathrm{一个}(X){你}^q=0\text{在 }\mathrm{\Omega },你=0\text{上 }\partial \mathrm{\Omega }.$$这里，$\mathrm{\Omega }\subset {ℝ}^{ñ}$是一个有界平滑域，$F$是一个完全非线性的椭圆算子，$\mathrm{一个}：\mathrm{\Omega }\to ℝ$是一个符号变化的权重，$\gamma \ge 0$， 和$0<q<\gamma +1$. 我们证明这个问题没有非平凡的死核解决方案，如果要么$q$足够接近$\gamma +1$或负面的部分$\mathrm{一个}$足够小。此外，我们在这些条件下获得了正解的存在性和唯一性$q$和$\mathrm{一个}$. 我们的结果扩展了之前在半线性情况下建立的结果，即使对于简单模型也是新的$|D你(X){|}^{\gamma }\mathrm{\text{Tr}}(\mathrm{\text{一个}}(X)D^2你(X))+\mathrm{一个}(X){你}^q(X)=0$， 在哪里$\mathrm{\text{一个}}\in C^0(\mathrm{\Omega };\mathrm{\text{符号}}(ñ))$是均匀椭圆非负矩阵。