Bifurcation from Zero or Infinity in Sturm–Liouville Problems Which Are Not Linearizable

AbstractWe consider the nonlinear Sturm–Liouville problem[formula][formula]whereai,biare real numbers with |ai|+|bi|>0,i=0,1, λ is a real parameter, and the functionspandaare strictly positive on [0,π]. Suppose that the nonlinearityhsatisfies a condition of the form[formula]as either |(ξ,η)|→0 or |(ξ,η)|→∞, for some constantsM0,M1. Then we show that there exist global continua of nontrivial solutions (λ,u) bifurcating fromu=0 or “u=∞,” respectively. These global continua have properties similar to those of the continua found in Rabonowitz' well-known global bifurcation theorem