论文标题
完全非线性椭圆模型中死核的不存在
Non-existence of dead cores in fully nonlinear elliptic models
论文作者
论文摘要
我们研究了问题$$ $$ | du |^γf（x，d^2u）+a（x）u^q = 0 \ quad \ mbox {in} \ quadω，\ quad U = 0 \ quad quad \ quad \ mbox {on} \ Mathbb {r}^n $是一个有界的平滑域，$ f $是一个完全非线性椭圆运算符，$ a：ω\ to \ mathbb {r} $是一个换人的重量，$γ\ geq 0 $，$ 0 <q <q <q <q <γ+1 $。我们表明，如果$ q $足够接近$γ+1 $或$ a $的负部分足够小，则此问题没有非平凡的死核解决方案。此外，在这些条件下，我们在$ q $和$ a $的情况下获得了积极解决方案的存在和独特性。我们的结果扩展了以前在半连续情况下建立的结果，即使对于简单型号$ | d u（x）|^γ\ Mathrm {tr}（\ Mathrm {a}（x）（x）d^2 u（x）） + a（x） + a（x）u^{q}（q）椭圆形和非负矩阵。
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 $Ω\subset \mathbb{R}^N$ is a bounded smooth domain, $F$ is a fully nonlinear elliptic operator, $a: Ω\to \mathbb{R}$ is a sign-changing weight, $γ\geq 0$, and $0<q<γ+1$. We show that this problem has no non-trivial dead core solutions if either $q$ is close enough to $γ+1$ 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 $|D u(x)|^γ \mathrm{Tr}(\mathrm{A}(x) D^2 u(x)) + a(x)u^{q}(x) = 0$, where $\mathrm{A} \in C^0(Ω;Sym(N))$ is a uniformly elliptic and non-negative matrix.
