A necessary and sufficient condition for existence of large solutions to semilinear elliptic equations

[摘要] We consider the semilinear equation Delta u = p(x)f(u) on a domain Omega subset of or equal to R-n, n greater than or equal to 3, where f is a nonnegative, nondecreasing continuous function which vanishes at the origin, and p is a nonnegative continuous function with the property that any zero of p is contained in a bounded domain in Omega such that p is positive on its boundary. For Omega bounded, we show that a nonnegative solution u satisfying u(x) --> infinity as x --> delta Omega exists if and only if the function psi(s) = integral(0)(s) f(t) dt satisfies integral(1)(infinity)(psi(s))(-1/2) ds < infinity. For Omega unbounded (including Omega = R-n), we show that a similar result holds where u(x)--> infinity as \ x \ --> infinity within Omega and u(x) --> infinity as x --> delta Omega if p(x) decays to zero rapidly as \ x \ --> infinity. (C) 1999 Academic Press.

[发布日期] 1999-12-01

[关键词] large solution;elliptic equation;semilinear equation