In this article we apply the minmax principle we developed in [6] to obtain sign-changing solutions for superlinear and asymptotically linear Dirichlet problems. We prove that, when isolated, the local degree of any solution given by this minmax principle is +1. By combining the results of [6] with the degree-theoretic results of Castro and Cossio in [5], in the case where the nonlinearity is asymptotically linear, we provide sucient conditions for: i) the existence of at least four solutions (one of which changes sign exactly once), ii) the existence of at leastve solutions (two ofwhichchangesign), andiii)theexistenceofpreciselytwosign-changing solutions. For a superlinear problem in thin annuli we prove: i) the existence of a non-radial sign-changing solution when the annulus is suciently thin, andii)theexistenceofarbitrarilymanysign-changingnon-radialsolutions when, in addition, the annulus is two dimensional. The reader is referred to [7] where the existence of non-radial signchanging solutions is established when the underlying region is a ball.