2011/10/11 by Rainer Mandel, Mandel, Rainer, Wolfgang Reichel +1
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP
paper · pdf · doi:10.48550/arxiv.1110.2314
arxiv created 2011/10/11 · arxiv updated 2011/10/12
We consider the nonlinear Schrödinger equation -Δu + V(x) u = Γ(x) |u|p-1u in \Rn where the spectrum of -Δ+V(x) is positive. In the case n≥ 3 we use variational methods to prove that for all p∈ ((n)/(n-2),(n)/(n-2)+\eps) there exist distributional solutions with a point singularity at the origin provided \eps>0 is sufficiently small and V,Γ are bounded on \Rn∖ B1(0) and satisfy suitable Hölder-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 -Δu + V(x) u = g(x,u) is a bounded strong solution if V is bounded and g satisfies certain growth conditions.