Odd linking and bifurcation in gaps: the weakly indefinite case

In this paper, we consider nonlinear Schrodinger equations of the following type: -Delta u(x) + V (x) u(x) -q(x)|u(x)|sigma u(x) =lambda u(x), x is an element of R-N, u is an element of H-1(R-N) \ {0}, where N >= 2 and sigma > 0. We concentrate on situations where the potential function V appearing in the linear part of the equation is of Coulomb type; by this we mean potentials where the spectrum of the linear operator -Delta + V consists of an increasing sequence of eigenvalues lambda(1), lambda(2), followed by an interval belonging to the essential spectrum. We study, for lambda kept fixed inside a spectral gap or below lambda(1), the existence of multiple solution pairs, as well as the bifurcation behaviour of these solutions when lambda approaches a point of the spectrum from the left-hand side. Our method proceeds by an analysis of critical points of the corresponding energy functional. To this end, we derive a new variational characterization of critical levels c(0)(lambda)