Using techniques of bifurcation theory we present two exact multiplicity results for boundary value problems of the typeThe first result concerns the case when the nonlinearity is independent of x and behaves like a cubic in u. The second one deals with a class of nonlinearities with explicit x dependence.

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SynopsisWe consider weakly-coupled elliptic systems of the typewith each fi being either an increasing or a decreasing function of each uj. Assuming the existence of coupled super- and subsolutions, we prove the existence of solutions, and provide a constructive iteration scheme to approximate the solutions. We then apply our results to study the steady-states of two-species interaction in the Volterra–Lotka model with diffusion.

Multiplicity results are provided for two classes of boundary-value problems with cubic nonlinearities, depending on a parameter A. In particular, it is proved that for sufficiently large A, there are exactly two solutions, and that all solutions lie on a single smooth solution curve. The last fact allows one to use continuation techniques to compute all solutions.