S -Shaped connected component for nonlinear fourth-order problem of elastic beam equation

We investigate the existence of s-shaped connected component in the set of positive solutions of the fourth-order boundary value problem: u′′′′x=hxfux,xe(0,1),u(0)=u(1)=u′′0=u′′1=0,where >0 is a parameter,hec[0,1],and fec[0,∞) with f0lims→0(f(s)/s)=∞. we develop a bifurcation approach to deal with this extreme situation by constructing a sequence of functions f[n] satisfying f[n]→f and (f[n])0→∞. by studying the auxiliary problems,we get a sequence of unbounded connected components c[n],and,then,we find an unbounded connected component c in the set of positive solutions of the fourth-order boundary value problem which satisfies 0,0c[n] and is s-shaped. © 2017 jinxiang wang et al.