Abstract
We obtain the critical exponent for the Dirichlet exterior problem
iu(t) + Delta u = lambda vertical bar u vertical bar(p) in (0,infinity) x D-c,
u(t,x) = f(x), in (0,infinity) x partial derivative D,
u(0,x) = g(x), in D-c,
where u is a complex valued function, D = (B(0,1) over bar is the closed unit ball in R-N, N >= 3, D-c is its complement, p > 1, lambda epsilon C\{0}, f not equivalent to 0, f epsilon L-1(partial derivative D; C) and g epsilon L-loc(1)(<(D-c)over bar>C). More precisely, we show that
. if 1 < p < p* := N/N-2 and
Re lambda . Im integral(Dc) g(x)h(x) dx < 0 and Re lambda . Re integral(partial derivative D)f (x)dS(x) < 0 or
Im lambda . Re integral(Dc)g(x)h(x) dx > 0 and Im lambda . Im integral(partial derivative D)f (x) dS(x) < 0,
where h is a certain harmonic function, then the considered problem possesses no global weak solution.
. If p > p*, then the problem admits global solutions for some lambda, f and g. (C) 2018 Elsevier Inc. All rights reserved.