Σ-shaped bifurcation curves

Abstract

We study positive solutions to the steady state reaction diffusion equation of the form: 'Equation Presented' where γ > 0 is a positive parameter, Ω is a bounded domain in ℝN when N > 1 (with smooth boundary ∂Ω) or Ω = (0, 1), and ∂u/∂η is the outward normal derivative of u. Here f(s) = ms + g(s) where m ≥ 0 (constant) and g ∈ C2[0, r) ∩ C[0, ∞) for some r > 0. Further, we assume that g is increasing, sublinear at infinity, g(0) = 0, g′(0) = 1 and g″(0) > 0. In particular, we discuss the existence of multiple positive solutions for certain ranges of γ leading to the occurrence of Σ-shaped bifurcation diagrams. We establish our multiplicity results via the method of sub-supersolutions.