Existence and multiplicity results for a class of semilinear elliptic equations

Abstract

We study the existence and multiplicity of nonnegative solutions, as well as the behaviour of corresponding parameter-dependent branches, to the equation $-\hat{I}”u=(1-u)u^m-\hat{I}»u^n$ in a bounded domain $\hat{I}\text{c◯}\subset {\mathbb{R}}^N$ endowed with the zero Dirichlet boundary data, where $0<m\le 1$ and $n>0$. When $\hat{I}»>0$, 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.

25 pages, 12 figures. Two figures and three additional references added according to referee's suggestions; several minor typos corrected. Published in Nonlinear Analysis