On the global structure of the set of positive solutions of some semilinear elliptic boundary value problems (Q1903053)

This paper analyzes the structure of the set of positive solutions of \(- \Delta u= \lambda u+ f(u)- u^{p+ 1}\) in \(\Omega\) with zero Dirichlet boundary condition. The global continuum of positive solutions coming from the trivial equilibrium at the principal eigenvalue of the linearization is constituted by a regular curve if the slope of the kinetic at the trivial solution is large enough and \(\Omega\) is convex (or simply connected). The width of the boundary layer of a singular perturbation is used to obtain the results.