On the Existence of Sign Changing Solutions for Semilinear Dirichlet Problems

where Ω ⊂ R is a bounded domain with Lipschitz boundary and f : R → R is of class C with f(0) = 0. Thus u0 ≡ 0 is a trivial solution of (D) and we are interested in finding and studying nontrivial solutions. One way of obtaining these is to compare the behavior of f near the origin and near infinity. We shall always assume that f grows subcritically at infinity so that variational methods can be applied and the associated functional satisfies the Palais–Smale condition. Suppose f ′(0) < λ1 where 0 < λ1 < λ2 ≤ λ3 ≤ . . . are the eigenvalues (counted with multiplicities) of −∆ on Ω with homogeneous Dirichlet boundary conditions. If f grows superlinearly at infinity then the mountain pass theorem of Ambrosetti and Rabinowitz [AR], [R] together with the maximum principle guarantees the existence of a positive solution u+ and a negative solution u− of (D). Using linking or Morse type arguments Wang [Wa] obtained a third nontrivial solution u1. In this paper we shall refine Wang’s result and obtain more information on u1 and on other solutions whose existence is proved via Morse theory. Let us illustrate this with the following two theorems. More general results will be stated and proved later.