Bifurcation from the essential spectrum without sign condition on the nonlinearity

We study a problem from nonlinear optics that leads to an integro-differential equation which can be written as the abstract bifurcation problem Au = lambda Lu + del phi (u). For such a type of equation, a general theorem on the bifurcation of solutions from the essential spectrum is proved. The hypothesis that phi is non-negative (as is usually assumed) could be omitted and replaced by other conditions that allow us to use the results for the particular example. We also indicate applications to semilinear elliptic equations.