Abstract

the real-valued functions a:’ = a$), a$, rz,) 2 0, 1 5 i, j 5 N, 1 I k 5 r are Lipschitz continuous on 0 and the matrix (a?‘), 1 I k 5 r, is positive definite on a. The operators Lk are uniformly elliptic. In the case of a single equation, r = 1, Ambrosetti and Hess, [2], have proved that the bifurcation from infinity occurs when the nonlinearity, f, is asymptotically linear and its asymptotic derivative positive. Peitgen and Schmitt, [8, 91, have obtained similar results for nonlinearities that satisfy more general asymptotic boundedness conditions. We extend these results to the system (1. l), r L 1. In Section 3 we prove that the bifurcation from infinity occurs at some A*, when f is asymptotically linear and its asymptotic derivative, M(x) = (m,(X)), satisfies the following conditions: mij 2 0, i # j, and for at least one i E [ 1, . . . , r), rn: # 0. (If we look at the linearized system (at co), these are the conditions that guarantee the existence of a positive eigenvalue with a positive eigenfunction, a result due to Hess, [5].) Our result is analogous to that of Hess, [5, theorem 21, treating the case of bifurcation from the trivial solution, and we obtain the same kind of estimates for A *. We use global perturbation techniques, employed by Peitgen and Schmitt, [8, 91, for the case of a single equation. This method enables us to also trace these bifurcations via global continua of solutions of a perturbed problem.

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