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 $-\mathrm{\Delta }u=(1-u)u^m-\lambda u^n$ in a bounded domain $\Omega \subset $\mathbb{R}$^N$endowedwiththezeroDirichletboundarydata,where$0<m \leq 1$and$n>0$\mathrm{.}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.