Bifurcation Problems for Generalized Beam Equations

We investigate a class of bifurcation problems for generalized beam equations and prove that the one-parameter family of problems have exactly two bifurcation points via a unified, elementary approach. The proof of the main results relies heavily on calculus facts rather than such complicated arguments as Lyapunov-Schmidt reduction technique or Morse index theory from nonlinear functional analysis.


Introduction and Main Results
In physics, the vibration of an elastic beam, with length 1 and one endpoint hinged at  = 0, which is compressed at the free edge ( = 1) by a force of intensity proportional to  > 0, is governed by the so-called beam equation   +  sin  = 0 in (0, 1) ; (1) see [1].The beam maintains its shape when the "force"  is sufficiently small, but it will buckle once  exceeds a certain value.In mathematics, the set of such values can be studied by exploiting the homogeneous Neumann boundary value problem: +  sin  = 0 in (0, 1) ,   (0) =   (1) = 0. ( Before stating precisely the properties which we will explore in BVP (2), we embed this problem into a family of such boundary value problems; that is, we introduce the family of problems   + ℎ ∘  = 0 in (, ) ,   () =   () = 0, where ,  ∈ R with  < ,  belongs to a certain nonempty subset of R, and  = () is the unknown; the function ℎ ∈  ∞ (R; R) satisfies the following: there exists an  > 0 such that (H1) ℎ( + ) = −ℎ(), for all  ∈ R, (H2)  0 ∈ R for which : [0, ] ∋   → √∫ Remark 1.We call the equation occurring in BVP (3) "generalized" beam equation; such equations are widely used to describe various physical phenomena.
Trivially, BVP (3) admits the trivial solution  = 0 for any  ∈ R.Here we are focused on the bifurcation theory for BVP (3).The bifurcation points are determined by eigenvalues associated with the differential operator   + ℎ ∘ .At such 2 Advances in Mathematical Physics points, the number of solutions to (3) may change.However, very little further work has been done to determine whether the number of solutions changes at these points.In this paper, we give such a criterion for a class of nonlinear problems.Theorem 4. Let , ,  ∈ R with  < .Assume (H1)-(H2).Then ± 2 (( − ) 2 ℎ  ( 0 )) (7) are two bifurcation points for BVP (3).Besides, (3) has nonconstant solutions if and only if The proof of a bifurcation assertion of a nonlinear equation often has as ingredients such topological arguments as Krasnoselskii's and Rabinowitz's theorems on bifurcation.These arguments usually have the assumption that the algebraic multiplicity of the associated linear eigenvalue problem is odd; see [1][2][3] and the references therein.Since then, several authors have also attempted to remove such oddness assumption; see [1,2,4].In particular, Ma and Wang [2] developed an elaborate algorithm to prove steady state bifurcation assertions concerning nonlinear equations; this algorithm does not assume the oddness of the algebraic multiplicity.See [5][6][7][8][9][10][11][12][13] for more studies on bifurcation problems.Our approach to prove Theorem 4 does not assume such parity condition on the algebraic multiplicity.
As a matter of fact, BVP (2) is a special case of Sturm-Liouville problem or boundary value problems for elliptic partial differential equations.Therefore, BVP (2), possibly in disguise, has been studied extensively in the literature for the existence of solutions satisfying certain prescribed properties, for qualitative properties of solutions, and so on; see [14][15][16][17] and the profound references cited therein.
The remainder of this paper is organized as follows.In Section 2 we introduce some nonlinear functional analysis and formulate the problem in a formal way, and in Section 3 we give the proof of Theorem 4.

The Existing Bifurcation Results for BVP (2)
In this section, we mainly give a brief review of the existing results in the literature concerning bifurcation problems for BVP (2) which can be viewed as archetypes of bifurcation problems for BVP (3).Indeed, bifurcation problems for BVP (2) have been often provided as illustration examples to test the proposed abstract bifurcation-problems-solving method in the literature; see [1,12,13,18].
In particular, Ma and Wang [18] proposed an abstract method which generalizes slightly the previous one obtained by Nirenberg [1].In presenting their method, the authors fixed two Banach spaces  and  for which  embeds continuously and densely into .The abstract problem which they were concerned with reads where   :  → ,  ∈ R, is a family of bounded linear operators and   :  →  is a family of continuous mappings.They assumed the following.
(H3)   is in the form   =  +   with  as a linear topological isomorphism of  onto  and   as compact linear operators; hence the spectrum of   consists of the exactly countably many eigenvalues {  ()} (listed by algebraic multiplicities) of   ; there exists  0 for which (H4) For any  > 0, there exists a  > 0 such that is analytic in the sense that is a continuous, symmetric -form on .
The precise problem with which they are concerned is whether there is a  0 ∈ R given in such a way that if   ̸ = 0 with  in a neighborhood of  0 is a collection of solutions to BVP (9), then If there exists a  0 which satisfies the above requirements, then  0 is called a bifurcation point for nonlinear problem (9); also, problem ( 9) is said to bifurcate from  0 .Concerning (9), they proved the following.Assume (H3)-(H4).Then  0 is a candidate bifurcation point of the nonlinear problem (9).
The proof of the above theorem provided in [18] utilizes such complicated methods as Lyapunov-Schmidt reduction method, Morse index theory, and so forth.
Here we are tempted to use the results obtained in Ma and Wang [18] to solve the bifurcation problem for BVP (3); it is however obvious that the nonlinear reaction R ∋  → ℎ(sin ) ∈ R precludes our application of such results.In the next section, we will analyze the bifurcation problem for BVP (3) in an elementary way.

Proof of the Main Results
In this section we propose two lemmas and then prove Theorem 4 based on them.Various calculus theorems are employed in our proofs, and the elementary equality is also used repeatedly.
For the other half, we assume without loss of generality that  > 0, and assume (8) holds; that is, And we show that (2) has a nonconstant solution.