Positive solutions to superlinear second–order divergence type elliptic equations in cone–like domains

We study the problem of the existence and nonexistence of positive solutions to a superlinear second–order divergence type elliptic equation with measurable coefficients −∇ · a · ∇u = u (∗), p > 1, in an unbounded cone–like domain G ⊂ R (N ≥ 3). We prove that the critical exponent p∗(a,G) = inf{p > 1 : (∗) has a positive supersolution at infinity in G } for a nontrivial cone– like domain is always in (1, N N−2 ) and in contrast with exterior domains depends both on the geometry of the domain G and the coefficients a of the equation.