LACUNARY BIFURCATION FOR OPERATOR-EQUATIONS AND NONLINEAR BOUNDARY-VALUE-PROBLEMS ON RN

We consider nonlinear eigenvalue problems of the form Lu + F(u) = lambda-u in a real Hilbert space, where L is a positive self-adjoint linear operator and F is a nonlinearity vanishing to higher order at u = 0. We suppose that there are gaps in the essential spectrum of L and use critical point theory for strongly indefinite functionals to derive conditions for the existence of non-zero solutions for lambda-belonging to such a gap, and for the bifurcation of such solutions from the line of trivial solutions at the boundary points of a gap. The abstract results are applied to the L2-theory of semilinear elliptic partial differential equations on R(N). We obtain existence results for the general case and bifurcation results for nonlinear perturbations of the periodic Schrodinger equation.