On the critical exponent for nonlinear Schrödinger equations without gauge invariance in exterior domains

We obtain the critical exponent for the Dirichlet exterior problemiut+Δu=λ|u|p,in(0,∞)×Dc,u(t,x)=f(x),in(0,∞)×∂D,u(0,x)=g(x),inDc, where u is a complex valued function, D=B(0,1)‾ is the closed unit ball in RN, N≥3, Dc is its complement, p>1, λ∈C\{0}, f≢0, f∈L1(∂D;C) and g∈Lloc1(Dc‾;C). More precisely, we show that