Positive solutions of semilinear elliptic problems in unbounded domains

Abstract

Di= a/ax,, i= l,..., N; each a;,E C,,, ’ + YQ), b E Cg,,(Q), h(x) 3 b, > 0, 0 < CI < 1; and f(x, U) satisfies assumptions (f,)-(fs) below. In particular it is assumed that f(x, 0) = 0 for all x E 52, implying that the boundary value problem (1.1) always has the trivial solution. Our main purpose is to establish the existence of a positive solution of (1.1) throughout 1;2 in cases for which the nonlinearity in (1.1) is unbounded above, i.e., f(x, r)/t -+ +co as t + +co locally uniformly in Q. We treat the case of bounded nonlinearities elsewhere [20]. The main Theorem 4.5