Sign changing solutions of superlinear Schrodinger equations

We prove the existence of sign changing solutions in H-l(R-N) for a stationary Schrodinger equation -Deltau + a(x)u = f(x, u) with superlinear and subcritical nonlinearity f, and control the number of nodal domains. Iff is odd we obtain an unbounded sequence of sign changing solutions u(k), k greater than or equal to 1, so that u(k) has at most k + l nodal domains. The bound on the number of nodal domains follows from a nonlinear version of Courant's nodal domain theorem which we also prove.