The Principle Branch of Solutions of a Nonlinear Elliptic Eigenvalue Problem onRN

We obtain existence, uniqueness and asymptotic decay properties of a semilinear elliptic eigenvalue problem inRN. The corresponding problem in dimensionN=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 (λ, u) bifurcating in the spacesR×C1(RN) orR×W2, p(RN) from the trivial solution atλ=Λ. HereΛ<0 is the lowest eigenvalue of the corresponding linear problem. We show that forplarger than a critical value depending onNthe branch is bounded inR×W2, p(RN), and for smallerpit is unbounded inR×Lp(RN). The unboundedness for smallpis 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. Forλ=0 solutions do not decay exponentially, and we prove uniqueness only forλ<0. No assumptions are made concerning the growth of the nonlinearity at ∞.