Σ-Shaped Bifurcation Curves

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.

$ \Sigma $-shaped bifurcation curves for classes of elliptic systems

“…Introduction. Recently, there has been growing interest on the study of the reaction diffusion models where a parameter influences the equation as well as the boundary condition (see [1], [2], [6], [8], [7], and [9]). In this paper, we study reaction diffusion systems of the form:…”

“…in [1], we obtained exact bifurcation curves via the quadrature method and Mathematica computations. See below the graph of f along with the corresponding bifurcation curves we obtained in [1] for two sets of parameter values (k, α). We present some preliminaries in Section 2.…”

“…First we construct sub-super solutions for certain λ ranges. Here we extend the ideas used in [1] appropriately for the systems case to construct sub-super solutions for (1.1) when f and g satisfy a combined sub-linear condition at infinity. Recall θ λ and σ λ (see (1.3)).…”

“…(a) Graphs of f for two sets of parameter values (k, α). (b) Exact bifurcation curves for single equation case for two sets of parameter values (k, α) (see[1]). …”