The Gelfand equation is a second order quasilinear elliptic differential equation arising in various models such as combustion, thermal explosions, gravitational equilibrium of polytropic stars and conformal geometry. In combustion theory the temperature satisfies a Gelfand equation in a fixed domain supplemented with boundary conditions of Dirichlet and Robin type.
The mathematics of the Gelfand equation has a long history starting with the discussion of the one-dimensional problem and the radial solutions. It received a significant upturn with the introduction of functional analysis. The nonlinearity depends on a parameter which is crucial for the existence of solutions. This parameter led to the study of nonlinear eigenvalue problems and the theories of bifurcation and stability came into play.

In this talk we give an overview of known results of nonlinear eigenvalue problems in the context of the Gelfand problem, mention some open problems and indicate possible further developments.