Σ-Shaped Bifurcation Curves

Abstract

Abstract We study positive solutions to the steady state reaction diffusion equation of the form: − Δ u = λ f ( u ) ;   Ω ∂ u ∂ η + λ u = 0 ;   ∂ Ω $$\begin{array}{} \displaystyle \left\lbrace \begin{matrix} -{\it\Delta} u =\lambda f(u);~ {\it\Omega} \\ \frac{\partial u}{\partial \eta}+ \sqrt{\lambda} u=0;~\partial {\it\Omega}\end{matrix} \right. \end{array}$$ where λ > 0 is a positive parameter, Ω is a bounded domain in ℝ N when N > 1 (with smooth boundary ∂ Ω) or Ω = (0, 1), and ∂ u ∂ η $\begin{array}{} \displaystyle \frac{\partial u}{\partial \eta} \end{array}$ is the outward normal derivative of u. Here f(s) = ms + g(s) where m ≥ 0 (constant) and g ∈ C 2[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 λ leading to the occurrence of Σ-shaped bifurcation diagrams. We establish our multiplicity results via the method of sub-supersolutions.