EXPLICIT ESTIMATES ON POSITIVE SUPERSOLUTIONS OF NONLINEAR ELLIPTIC EQUATIONS AND APPLICATIONS

摘要

In this paper we consider positive supersolutions of the nonlinear elliptic equation-Delta u = rho(x) f(u) vertical bar del u vertical bar(p), in Omega,Where 0 = p 1, Omega is an arbitrary domain (bounded or unbounded) in R-N (N = 2), f : [0, a(f)) - R+ (0 a(f) = +infinity) is a non-decreasing continuous function and rho: Omega - R is a positive function. Using the maximum principle we give explicit estimates on positive supersolutions u at each point x is an element of Omega where del(u) not equivalent to 0 in a neighborhood of x. As applications, we discuss the dead core set of supersolutions on bounded domains, and also obtain Liouville type results in unbounded domains Omega with the property that sup(x is an element of Omega) dist(x, partial derivative Omega) = infinity. In particular when rho(x) =vertical bar x vertical bar(beta) (beta is an element of R) and f(u) = u(q) with q + p 1 then every positive supersolution in an exterior domain is eventually constant if(N - 2)q + p(N - 1) N + beta.