We study the existence and multiplicity of nonnegative solutions, as well as the behavior of corresponding parameter-dependent branches, to the equation $-\Delta u=(1-u)u^m-\lambda u^n$ in a bounded domain $\Omega \subset {\mathbb{R}}^N$ endowed with the zero Dirichlet boundary data, where $0<m\le 1$ and $n>0$. When $\lambda >0$, the obtained solutions can be seen as steady states of the corresponding reaction–diffusion equation describing a model of isothermal autocatalytic chemical reaction with termination. In addition to the main new results, we formulate a few relevant conjectures.

中文翻译：

---

一类半线性椭圆方程的存在性和多重性结果

我们研究方程的非负解的存在性和多重性以及相应的依赖参数的分支的行为 $-\Delta ü=（\mathrm{1个}-ü）{ü}^{米}-\lambda {ü}^{ñ}$ 在有界域中 $\Omega \subset {\mathbb{[R}}^{ñ}$ 赋予Dirichlet边界数据为零，其中 $0<米\le \mathrm{1个}$ 和 $ñ>0$。什么时候$\lambda >0$，获得的溶液可以看作是相应的反应扩散方程的稳态，描述了等温自催化终止化学反应的模型。除了主要的新结果外，我们还提出了一些相关的猜想。