Existence and multiplicity results for a class of semilinear elliptic equations

We study the existence and multiplicity of nonnegative solutions, as well as the behaviour 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 $0<m \leq 1$ 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.