Lacunary bifurcation for operator equations and nonlinear boundary value problems on ℝN

Abstract

SynopsisWe consider nonlinear eigenvalue problems of the form Lu + F(u) = λu 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 λ 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 ℝN. We obtain existence results for the general case and bifurcation results for nonlinear perturbations of the periodic Schrödinger equation.