A Superlinar Sturm-Liouville Eigenvalue Problem

In this paper, we consider the Sturm-Liouville eigenvalue problem: BVP(1.1) where R1(u(?))=?u(?)??p(?)u?(?), R2(u(1))=?u(1)??p(1)u?(1), f(x,u) is continuous and satisfies a local Lipschitz condition with respect to u ,and for each fixed x , f(x,u) is odd , f(x,?)??,uf(x,u)> ?,u??. When f(x,u) is sublinear with respect to u, G.H.Pimbley ,by using a technique of the phase plane , obtained a clear global structure of all the solutions of BVP(1.1) in 1962 (see [1]).After the 1980's , many interesting results about superlinear boundary value problems were obtained by using topological degree or coincidence degree theory (Sometimes time-map technique was employed for autonomous problems)(see [2]-[7]).Perhaps because of the limitations of the topological degree theory ,a global structure of all the solutions of the superlinear problems failed to be given. Inspired by G.H.Pimbley's paper [1],we use the technique of the phase plane to approach the global structure of the superlinear problem BVP(1.1).The main results obtained is the following.Assume limu?? f(x,u)?u=?(x),limu?? f(x,u)?u=?(x),fu??,ufuu>? for u??.Let ??n?and ??n? be the sequences of eigenvalues for the boundary value problems and respectly , then we haveTheorem 1 If the above conditions hold,then BVP(1.1)'s spectrum includes ??n??n?(n ?1). For each ????n??n ?, BVP(1.1) has at least two nontrivial solutions with contrary sign, say un?x??? and ?un?x???, which have n-1 zeros in (?,1).Especially for each ??(?1??1?,BVP(1.1) has one positive solution and one negative solution. Further more, we have?? un?x??? ??C[??????, ???nï¼› ?? un?x??? ??C[??????, ???n. And for each m?(?,??, there exists a ????n ??n? such that the solution un?x??? related to the ? satisfies ?? un?x??? ??C[????=m. ?=?n is a bifurcation point at which ?? un?x??? ??C[????=?.If p(x)??,f(x,u)=f(u),that is to say ,BVP(1.1) is a autonomous problem ,then "at least" and "includes" in theorem 1 will be replaced respectly by "exactly" and "is consisted of ".Our results give a clear global structure of all the solutions of the autonomous superlinear eigenvalue problem and a relatively clear global structure for BVP(1.1).The theorem naturally implies all the results in [2]-[7].At the end, we have to point out, with great pity, that whether BVP(1.1) has exactly two solutions with n-1 zeros in (?,1) for ????n??n ? and whether BVP(1.1) has other solutions for ??[?n??n ] are still left for further study.