Bifurcation into spectral gaps for strongly indefinite Choquard equations

In this paper, we consider the semilinear elliptic equations(-delta u +V(x)u= (I-alpha * |u|(p))|u|(p-2)u + lambda u for x is an element of R-N,u(x) -> 0 as |x| -> infinity,where I alpha is a Riesz potential, p is an element of (N+alpha/N, N+alpha/N-2 ), N >= 3 and V is continuous periodic. We assume that 0 lies in the spectral gap (a, b) of -delta + V. We prove the existence of infinitely many geometrically distinct solutions in H-1(R-N) for each lambda is an element of (a, b), which bifurcate from b if N+alpha/N < p < 1+2+alpha/N. Moreover, b is the unique gap-bifurcation point (from zero) in [a,b]. When lambda=a, we find infinitely many geometrically distinct solutions in H-loc(2)(R-N). Final remarks are given about the eventual occurrence of a bifurcation from infinity in lambda = a.