Sigma-Shaped Bifurcation Curves

We study positive solutions to the steady state reaction diffusion equation of the form:{-Delta u = lambda f(u); Omegapartial derivative u/partial derivative eta + root lambda u = 0; partial derivative Omegaan where lambda > 0 is a positive parameter, Omega is a bounded domain in R-N when N > 1 (with smooth boundary partial derivative Omega) or Omega = (0, 1), and partial derivative u/partial derivative eta is the outward normal derivative of u. Here f(s) = ms + g(s) where m >= 0 (constant) and g is an element of C-2 [0, r) boolean AND C[0, infinity) 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 A leading to the occurrence of Sigma-shaped bifurcation diagrams. We establish our multiplicity results via the method of sub-supersolutions.