论文标题
过去和最近对无限sublinear椭圆形问题的贡献
Past and recent contributions to indefinite sublinear elliptic problems
论文作者
论文摘要
我们在平滑的有界域$ - ω\ subset \ subset \ mathbb {r}^{n} $中，在平滑的有限域$-Δu= a（x）u^{q} $中审查无限的sublinear椭圆方程$-ΔU= a（x）u^{q} $，带有dirichlet或neumann均质边界条件。在这里$ 0 <q <1 $，$ a $是连续的，并且更改标志，在这种情况下，强大的最大原则不适用。结果，这些问题的非负解决方案集具有丰富的结构，尤其是死亡核心和/或积极的解决方案。总体而言，我们对积极解决方案的存在$ a $ a和$ q $的足够和必要的条件感兴趣。我们描述了过去几十年的主要结果，并将其与我们最近的贡献相结合。简要概述了证明。
We review the indefinite sublinear elliptic equation $-Δu=a(x)u^{q}$ in a smooth bounded domain $Ω\subset\mathbb{R}^{N}$, with Dirichlet or Neumann homogeneous boundary conditions. Here $0<q<1$ and $a$ is continuous and changes sign, in which case the strong maximum principle does not apply. As a consequence, the set of nonnegative solutions of these problems has a rich structure, featuring in particular both dead core and/or positive solutions. Overall, we are interested in sufficient and necessary conditions on $a$ and $q$ for the existence of positive solutions. We describe the main results from the past decades, and combine it with our recent contributions. The proofs are briefly sketched.
