S-Shaped Connected Component for Nonlinear Fourth-Order Problem of Elastic Beam Equation

We investigate the existence of -shaped connected component in the set of positive solutions of the fourth-order boundary value problem: , , where is a parameter, , and with . We develop a bifurcation approach to deal with this extreme situation by constructing a sequence of functions satisfying and By studying the auxiliary problems, we get a sequence of unbounded connected components , and, then, we find an unbounded connected component in the set of positive solutions of the fourth-order boundary value problem which satisfies and is -shaped.

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