We obtain existence, uniqueness and asymptotic decay properties of a semilinear elliptic eigenvalue problem in R N . The corresponding problem in dimension N=1, which provided the motivation for this work, leads to bounds for the wavelengths and the power of guided waves propagating in a medium consisting of layers of dielectric material whose refractive index depends on the intensity of the electric field. In this paper we show the existence of a continuous branch of solutions (*, u) bifurcating in the spaces R_C 1 (R N ) or R_W 2, p (R N ) from the trivial solution at *=4. Here 4<0 is the lowest eigenvalue of the corresponding linear problem. We show that for p larger than a critical value depending on N the branch is bounded in R_W 2, p (R N ), and for smaller p it is unbounded in R_L p (R N ). The unboundedness for small p is demonstrated by comparison with a radially symmetric problem. Decay estimates are obtained from explicitly constructed supersolutions having known asymptotic decay rates. Subsolutions can be obtained as small multiples of the eigenfunction of the linear problem. For *=0 solutions do not decay exponentially, and we prove uniqueness only for *<0. No assumptions are made concerning the growth of the nonlinearity at .