- Research Article
- Open Access
Nonlinear Systems of Second-Order ODEs
© P. Cerda and P. Ubilla. 2008
- Received: 2 February 2007
- Accepted: 16 November 2007
- Published: 10 December 2007
We study existence of positive solutions of the nonlinear system in ; in ; , where and . Here, it is assumed that , are nonnegative continuous functions, , are positive continuous functions, , , and that the nonlinearities satisfy superlinear hypotheses at zero and . The existence of solutions will be obtained using a combination among the method of truncation, a priori bounded and Krasnosel'skii well-known result on fixed point indices in cones. The main contribution here is that we provide a treatment to the above system considering differential operators with nonlinear coefficients. Observe that these coefficients may not necessarily be bounded from below by a positive bound which is independent of and .
- Partial Differential Equation
- Ordinary Differential Equation
- Nonlinear System
- Functional Equation
- General System
We study existence of positive solutions for the following nonlinear system of second-order ordinary differential equations:
(H1) We have
(H4) We have
Here are some comments on the above hypotheses. Hypothesis (H1) is a superlinear condition at 0 and Hypothesis (H2) is a local superlinear condition at . About hypothesis (H3), the fact that , are unbounded leads us to use the strategy of considering a truncation system. Note that if , are bounded, we would not need to use that system. Hypothesis (H4) allows us to have a control on the nonlinear operator in system 1.1.
We remark that, the case when and , systems of type (1.1) have been extensively studied in the literature under different sets of conditions on the nonlinearities. For instance, assuming superlinear hypothesis, many authors have obtained multiplicity of solutions with applications to elliptic systems in annular domains. For homogeneous Dirichlet boundary conditions, see de Figueiredo and Ubilla , Conti et al. , Dunninger and Wang [3, 4] and Wang . For nonhomogeneous Dirichlet boundary conditions, see Lee  and Marcos do Ó et al. . Our main goal is to study systems of type (1.1) by considering local superlinear assumptions at and global superlinear at zero.
The main result is the following.
Assume hypotheses (H1) through (H4). Then system (1.1) has at least one positive solution.
One of the main difficulties here lies in the facts that the coefficients of the differential operators of System (1.1) are nonlinear and that they may not necessarily be bounded from below by a positive bound which is independent of and In order to overcome these difficulties, we introduce a truncation of system (1.1) depending on so that the new coefficient of the truncation system becomes bounded from below by a uniformly positive constant. (See (2.2).) This allows us to use a fixed point argument for the truncation system. Finally, we show the main result proving that, for sufficiently large, the solutions of the truncation system are solutions of system (1.1). Observe that, in general, this system has a nonvariational structure.
The paper is organized as follows. In Section 2, we obtain the a priori bounds for the truncation system. In Section 3, we show that the a priori bounds imply a nonexistence result for system (2.4). In Section 4, we introduce a operator of fixed point in cones. In Section 5, we show the existence of positive solutions of the truncation system. In Section 6, we prove the main result, that is, we show the existence of a solution of system (1.1). Finally, in Section 7 we give some remarks.
In the next section, we will prove the existence of a positive solution for the following truncation system:
It is not difficult to prove that every solution of system (2.4) satisfies
In order to establish the a priori bound result we need the following two lemmas
Assume hypotheses (H2) and (H3). Then every solution of system (2.4) satisfies
we have that (2.7) is proved .
Assume hypotheses (H2) and (H3). Then Green's functions satisfy
which proves (2.12).
In this section, we see that the a priori bounds imply a nonexistence result for system (2.4).
which proves Theorem 3.1.
Consider the following Banach space:
In Section 5, we will give an existence result of the truncation system (2.3). The proof will be based on the following well-known fixed point result due to Krasnoselskis, which we state without proof (compare [8, 9]).
The following is an existence result of the truncation system.
Assume hipotheses (H1) through (H3). Then there exists a positive solution of system (2.3).
On the other hand, we have
which verifies (a) of Lemma 4.2. By Theorem 2.3, we clearly have (c).
The proof of Theorem 1.1 is direct consequence of the following.
A little modification of our argument may be done to obtain an existence result of the following more general system:
The authors are supported by FONDECYT, Grant no. 1040990.
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