- Research Article
- Open Access
Nontrivial Solutions of the Asymmetric Beam System with Jumping Nonlinear Terms
© The Author(s) Tacksun Jung and Q-Heung Choi. 2010
- Received: 8 October 2009
- Accepted: 11 September 2010
- Published: 21 September 2010
- Dirichlet Boundary Condition
- Nontrivial Solution
- Trivial Solution
- Suspension Bridge
- Critical Point Theory
where the nonlinearity crosses an eigenvalue. This equation represents a bending beam supported by cables under a load The constant represents the restoring force if the cables stretch. The nonlinearity models the fact that cables resist expansion but do not resist compression.
can furnish a model to study travelling waves in suspension bridges. This is a one-dimensional beam equation that represents only the up and down travelling waves of the beam. But the beam has also the right and left travelling waves. Hence we can consider two-dimensional beam equation (1.1).
The nonlinear equation with jumping nonlinearity has been extensively studied by many authors. For the fourth order elliptic equation, Taratello  and Micheletti and Pistoia [4, 5] proved the existence of nontrivial solutions, by using degree theory and critical point theory, separately. For one-dimensional case, Lazer and McKenna  proved the existence of nontrivial solution by the global bifurcation method. For this jumping nonlinearity, we are interested in the multiple nontrivial solutions of the equation. Here we used variational reduction method to find the nontrivial solutions of problem (1.1).
where , and are constants. This equation satisfies Dirichlet boundary condition on the interval and periodic condition on the variable . We use the variational reduction method to apply mountain pass theorem in order to get the main result that for (1.2) has at least three periodic solutions, two of which are nontrivial. In Section 5, we investigate the existence of multiple nontrivial solutions for perturbations of beam system (1.1). We also prove that for (1.1) has only the trivial solution.
Define . Then we have the following lemma (cf. ).
So the right-hand side of (2.10) defines a Lipschitz mapping of into with Lipschitz constant . Therefore, by the contraction mapping principle, there exists a unique solution . Since is a solution of (2.10), is the unique solution.
The following deformation theorem is stated in .
Let be a real Banach space and . Suppose satisfies Palais-Smale condition. Let be a given neighborhood of the set of the critical points of at a given level . Then there exists , as small as we want, and a deformation such that we denote by the set :
We state the Mountain Pass Theorem.
Lemma 4.1 (cf. ).
We will use a variational reduction method to apply the mountain pass theorem.
the right-hand side of (4.5) defines a Lipschitz mapping because for fixed maps into itself. By the contraction mapping principle, there exists a unique (also ) for fixed . Since is bounded from to there exists a unique solution of (4.4) for given .
is invariant under and under the map . So the spectrum of restricted to contains in . The spectrum of restricted to contains in . From the symmetry theorem in , any solution of this equation satisfies . This nontrivial periodic solution is periodic with periodic . This shows that there is no nontrivial solution of
where . Let us define the reduced functional on by . We note that we can obtain the same results as Lemmas 4.1 and 4.2 when we replace and by and . We also note that, for has only the critical point .
The proof of the lemma can be found in .
By Lemma 4.2, is Lipschitz continuous on . So the sequence is bounded in . Since ( ), it follows that and are bounded in . Since is a compact operator, there is a subsequence of converging to some in , denoted by itself. Since is a two-dimensional space, assume that sequence converges to with . Therefore, we can get that the sequence converges to an element in .
Now we state the main result in this paper.
two of which are nontrivial solutions.
We remark that is the trivial solution of problem (1.4). Then is a critical point of functional . Next we want to find others critical points of which are corresponding to the solutions of problem (1.4).
which is a contradiction. Therefore, there exists a critical point of at level such that , 0, which means that (1.4) has at least three critical points. Since , these two critical points coincide with two nontrivial period solutions of problem (1.4).
where and . When , from the above equation, we get the trivial solution . When , from the above equation, we get the nontrivial solutions , .Therefore, system(5.1) has at least three solutions , two of which are nontrivial solutions.
This work (Choi) was supported by Inha University Research Grant. The authors appreciate very much the referee's corrections and revisions.
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