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MULTIPLICITY RESULTS FOR A CLASS OF ASYMMETRIC WEAKLY COUPLED SYSTEMS OF SECOND-ORDER ORDINARY

MULTIPLICITY RESULTS FOR A CLASS OF ASYMMETRIC WEAKLY COUPLED SYSTEMS OF SECOND-ORDER ORDINARY DIFFERENTIAL EQUATIONS FRANCESCA DALBONO AND P. J. MCKENNA Received 1 November 2004 We prove the existence and multiplicity of solutions to a two-point boundary value problem associated to a weakly coupled system of asymmetric second-order equations. Applying a classical change of variables, we transform the initial problem into an equivalent problem whose solutions can be characterized by their nodal properties. The proof is developed in the framework of the shooting methods and it is based on some estimates on the rotation numbers associated to each component of the solutions. | MULTIPLICITY RESULTS FOR A CLASS OF ASYMMETRIC WEAKLY COUPLED SYSTEMS OF SECOND-ORDER ORDINARY DIFFERENTIAL EQUATIONS FRANCESCA DALBONO AND P J. MCKENNA Received 1 November 2004 We prove the existence and multiplicity of solutions to a two-point boundary value problem associated to a weakly coupled system of asymmetric second-order equations. Applying a classical change of variables we transform the initial problem into an equivalent problem whose solutions can be characterized by their nodal properties. The proof is developed in the framework of the shooting methods and it is based on some estimates on the rotation numbers associated to each component of the solutions to the equivalent system. 1. Introduction This paper represents a first step in the direction of extending to systems some of the well-known results established over the last two decades on nonlinear equations with an asymmetric nonlinearity. Recall that we call a nonlinearity asymmetric if the limits f to and f -to are different. The large literature on this type of nonlinear boundary value problem can be roughly summarized in the following statement in an asymmetric nonlinear boundary value problem with a large positive loading the greater the asymmetry the larger the number of multiple solutions. This principle applies in both the ordinary differential equation and partial differential equation setting and has significant implications for vibrations in bridges and ships. To illustrate the principle we consider the scalar problem u bu sin x u 0 u n 0 1.1 where we recall that u max u 0 u- max -u 0 . A combination of the results of 8 27 shows that if n2 b n 1 2 the problem 1.1 has exactly 2n solutions. Thus the greater the difference between f to namely b and f -to namely 0 the larger the number of solutions namely 2n . We sometimes say that the nonlinearity crosses the first n eigenvalues. Problem 1.1 has been widely studied in the literature. In addition to the papers 8 27 other contributions in .