Existence of Solutions for a Coupled System of Second and Fourth Order Elliptic Equations

Abstract

The generalized quasilinearization technique is applied to obtain a monotone sequence of iterates converging uniformly and quadratically to a solution of a coupled system of second and fourth order elliptic equations.

1. Introduction

The object of this paper is to study the existence of solutions of the following elliptic system: This problem is related to a coupled system of second and fourth ordinary differential equations: which is yielded from the steady state of the following Lazer-McKenna suspension bridge models proposed in [1]: Because of the important background, in [2], Ru and An presented the existence of single and multiple positive solutions of the 2p-order and 2q-order nonlinear ordinary differential systems by using the fixed-point theorem of cone expansion and compression type due to Krasnosel'skill; in [3], the authors studied the existence, multiplicity, and nonexistence of positive solutions for 2p-order and 2q-order systems of singular boundary value problems with integral boundary conditions; in [4], the author studied the existence, nonexistence, and multiplicity of solutions of an Ambrosetti-Prodi type problem for a system of second and fourth order ordinary differential equations by the variational method. In addition, in [5], An dealt with the maximum principle for the coupled system of second and fourth order elliptic equations: where the functions , are supposed to be nonnegative such that -matrix is cooperative. In addition, using the fixed point theorem, the author obtained the existence of solution for the following system:

Inspired by the above references, the aim of this paper is to study the existence of solutions for the elliptic system (1) by using the method of upper and lower solutions and the generalized quasilinearization technique on the basis of the maximum principle for the linear elliptic system (4) built by An in [5]. The quasilinearization method is pioneered by Bellaman and Kalaba [6] and generalized by Lakshmikantham and Vatsala [7]. The quasilinearization method has been applied to a variety of problems. We refer the readers to [8–12].

This paper is organized as follows. In Section 2, we give some preliminaries. In Section 3, we obtain a monotone sequence of approximate solutions converging uniformly and quadratically to a solution of (1).

2. Preliminaries

Let be a bounded smooth domain. denotes the Hilbert space with the inner production . The corresponding norm is ; denotes the Hilbert space with the inner production . The corresponding norm is .

Let be the positive first eigenvalue of the second order eigenvalue problem Then is the positive first eigenvalue of the fourth order eigenvalue problem From the Poincare inequality, for all , it follows that where .

Lemma 1 (see [5]). Assume that the cooperative matrix with , verifies condition Then (4) satisfies the maximum principle. By a maximum principle one means that if , then (4) has a nonnegative solution .

Lemma 2. Let , . Then (4) has a solution . In addition, there exists a constant such that

Proof. Let Ł denote the operator matrix and let denote the coefficient matrix Thus, (4) can be rewritten as where . From Lemma 1 it follows that the operator matrix is reversible, and is a compact positive operator. So . Furthermore, we can get for some constant .

3. Main Result

Letting , then (1) is equivalent to We introduce the following relation: which is ordered by the usual positive cone , where denotes the Banach space .

Definition 3. A function is a lower solution of (16) if it satisfies

Definition 4. A function is an upper solution of (16) if it satisfies
For convenience, we give some assumptions as follows.(A1)Equation (16) has a pair of lower and upper solutions , with .(A2), is quasimonotone nondecreasing in and is convex in uniformly in , exists and is continuous on , and is nondecreasing in for .(A3)The assumptions of Lemma 1 hold for with , .(A4), for some positive matrix .

Theorem 5. Assume that (A1)–(A4) hold. Then there exist two monotone sequences , , which converge uniformly to the solution of (1) or (16) in and the convergence is quadratic.

Proof. By (A2), we have where , . Define a function as follows: where , . It is clear to see that is quasimonotone nondecreasing in for each .
From Lemma 1 and (A3), it follows that the linear systems respectively, have a solution , .
Let . Then, from (22), we have Using Lemma 1 and (A3), it implies ; namely, .
Let . Then, from (23), we get Using Lemma 1 and (A3), the above inequality implies ; namely, .
Finally, let . Then, using (20), (22), and (23), we can obtain where . Using Lemma 1 and (A3), it yields that ; namely, . So, from the above discussions, we can get .
Continuing this process successively, then we can obtain two monotone sequences , satisfying where , are, respectively, the solutions of the following systems:
From (27), it is clear to see that , are uniformly bounded in . Moreover, is nondecreasing sequence and therefore there exists a subsequence which converges uniformly to . By the dominated convergence, it is easy to verify that is a solution of (1) in the distribution sense. Similarly, convergence holds for .
Now we show that the convergence of and is quadratic. For this purpose, let , . Then, using the mean value theorem, we have By (A3) and (A4), we get the inequality where .
Let be the unique solution of the linear problem From Lemma 1, (30), and (31), we know that . Furthermore, we have . From Lemma 2, we know that . Furthermore, we can obtain that .
In the similar way, we also have Therefore, the proof is complete.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This research is supported by the Natural Science Foundation of Jiangsu Province (Grant no. 1014-51314411).