Critical point theory on partially ordered Hilbert spaces

We develop some abstract critical point theory in order to prove that boundary value problems like the model problem {-Deltau = lambdau + \u \ (p-2)u in Omega {u = 0 on partial derivative Omega on a bounded domain Omega subset of R-N, 2 < p < 2N \ (N-2) have infinitely many sign changing solutions +/-u(k), k is an element of N, which are not comparable, that is, u(k) - u(l) and u(k) + u(l) change sign for k not equal l. We also show that there are no subsolutions u such that U < Uk for some k and u is positive somewhere. A corresponding nonexistence result applies to supersolutions, Related results on the existence of sign-changing solutions hold for other classes of nonlinearities. <(c)> 2001 Academic Press.