Second-order semilinear elliptic inequalities in exterior domains

Kondratiev Y, Liskevich V, Sobol Z. Second-order semilinear elliptic inequalities in exterior domains. JOURNAL OF DIFFERENTIAL EQUATIONS. 2003;187(2):429-455.We study the problem of existence and nonexistence of positive solutions of the semilinear elliptic inequalities in divergence form with measurable coefficients -del . a . delu + Vu - Wu(p)greater than or equal to0 in exterior domains in R-N, Ngreater than or equal to3. For W(x)asymptotic to\x\(-sigma) (sigmais an element ofR) at infinity we compute the critical line on the plane (p, a), which separates the domains of existence and nonexistence, and reveal the class of potentials V that preserves the critical line. Example are provided showing that the class of potentials is maximal possible, in certain sense. The case of (p, sigma) on the critical line has also been studied. (C) 2002 Elsevier Science (USA). All rights reserved