Distributional Solutions of the Stationary Nonlinear Schrodinger Equation: Singularities, Regularity and Exponential Decay

We consider the nonlinear Schrodinger equation -Delta u + V(x)u = Gamma(x)vertical bar u vertical bar(p-1)u in R-n where the spectrum of -Delta + V(x) is positive. In the case n >= 3 we use variational methods to prove that for all p is an element of (n/n-2, n/n-2 + epsilon) there exist distributional solutions with a point singularity at the origin provided epsilon > 0 is sufficiently small and V, Gamma are bounded on R-n \ B-1(0) and satisfy suitable Holder-type conditions at the origin. In the case n = 1, 2 or n >= 3, 1 < p < n/n-2, however, we show that every distributional solution of the more general equation -Delta u + V(x)u = g(x, u) is a bounded strong solution if V is bounded and g satisfies certain growth conditions.