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
We investigate the existence of multiple nontrivial solutions for perturbations and of the beam system with Dirichlet boundary condition in , in , where , and are nonzero constants. Here is the beam operator in , and the nonlinearity crosses the eigenvalues of the beam operator.
where and the nonlinearity crosses the eigenvalues of the beam operator. This system represents a bending beam supported by cables in the two directions.
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.
Since for all , we have that
() , where denotes the norm of ;
() if and only if .
Define . Then we have the following lemma (cf. ).
has only the trivial solution in .
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.
3. Mountain Pass Theorem
The mountain pass theorem concerns itself with proving the existence of critical points of functional which satisfy the Palais-Smale (PS) condition, which occurs repeatedly in critical point theory.
We say that satisfies the Palais-Smale condition if any sequence for which is bounded and as possesses a convergent sequence.
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.
Let be a real Banach space and satisfy condition. Suppose that
() there are constants such that , and
() there is an such that .
4. Critical Point Theory and Multiple Nontrivial Solutions
Then the functional is well defined in and the solutions of (1.4) coincide with the critical points of . Now we investigate the property of functional .
Lemma 4.1 (cf. ).
We will use a variational reduction method to apply the mountain pass theorem.
where , .
Let . Then satisfies a uniform Lipschitz continuous on with respect to the norm (also the norm ).
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 .
Therefore, is continuous on with respect to norm (also, to ).
If is a critical point of , then is a solution of (1.4) and conversely every solution of (1.4) is of this form.
Therefore, is strictly convex with respect to the second variable.
with equality if and only if .
Suppose that there exists such that . From (4.24), it follows that for all . Then by Lemma 4.2, it follows that for any . Therefore, is a solution of (1.4).
Conversely, if is a solution of (1.4) and , then for any .
Let , and . Then there exists a small open neighborhood of 0 in such that is a strict local minimum of .
where . It follows that is a strict local point of minimum of .
If , then the equation admits only the trivial solution in .
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
Let and . Then the functional , defined on , satisfies the Palais-Smale condition.
Let be a Palais-Smale sequence that is is bounded and in . Since is two-dimensional, it is enough to prove that is bounded in .
has only the trivial solution by Proposition 4.5. Hence is bounded in .
The above equation ( ) has only the trivial solution and hence has only one critical point .
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 .
Let , , and . Then we have for all with .
The proof of the lemma can be found in .
for all (certainly for also the norm ).
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 .
Since , this contradicts to the fact that for all . This proves that .
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).
If , then there exists a critical point of at level such that , 0 ( since and ). Therefore, in case , the functional has also at least 3 critical points .
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).
5. Nontrivial Solutions for the Beam System
where and the nonlinearity crosses the eigenvalues of the beam operator.
Let , , and . Then beam system (5.1) has at least three solutions , two of which are nontrivial solutions.
where the nonlinearity
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.
Let , and . Then system (5.1) has only the trivial solution .
where the nonlinearity
From (5.9), we get the trivial solution .
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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