Positive solutions of a class of nonlinear elliptic eigenvalue problems

This paper is mainly concerned with the natural order relationship between positive solutions of the elliptic eigenvalue Dirichlet problem: - Deltau = lambdaf (u) in Omega and u = 0 on partial derivativeOmega. Under suitable conditions, we prove that there are 2m - 1 positive solutions satisfying (u) over cap (1) < u(2)(*) < (u) over cap (2) < ... < u(m)(*) < (u) over cap (m). It seems that standard arguments do not provide such a result. Several authors, including P. Hess, proved the existence of equal number of positive solutions without such a relationship between them. We also prove that in Hess's result as well as in ours some sufficient condition is also necessary if the domain possesses a particular shape. At last, as an illustrative example, we study the diagram of positive solutions when lambdaf (u) = lambda(d + cos u) with lambda and d being both parameters.