EXISTENCE OF POSITIVE SOLUTIONS FOR NONLINEAR BOUNDARY-VALUE PROBLEMS IN UNBOUNDED DOMAINS OF Rn

Let D be an unbounded domain in R-n (n >= 2) with a nonempty compact boundary partial derivative D. We consider the following nonlinear elliptic problem, in the sense of distributions, Delta u = f(.,u), u > 0 in D, u vertical bar(partial derivative D) = alpha phi, lim(vertical bar x vertical bar ->+infinity)u(x)/h(x) = beta lambda, where alpha,beta,lambda areare nonnegative constants with alpha + beta > 0 and phi is a nontrivial nonnegative continuous function on partial derivative D. The function f is nonnegative and satisfies some appropriate conditions related to a Kato class of functions, and h is a fixed harmonic function in D, continuous on (D) over bar. Our aim is to prove the existence of positive continuous solutions bounded below by a harmonic function. For this aim we use the Schauder fixed-point argument and a potential theory approach.