Monotone solutions to a class of elliptic and diffusion equations

Abstract

In this paper, we study the existence and properties of monotone solutions to the following elliptic equation in Rn -Δu = F′(u), in Rn, ∂xn u > 0, and the diffusion equation u t - Δu = F′(u), in Rn × {t > 0}, ∂xn u > 0, u|t=0 = u 0, where A is the standard Laplacian operator in Rn, and u0 is a given smooth function in R" with some monotonicity condition. We show that under a natural condition on the nonlinear term F′, there exists a global solution to the diffusion problem above, and as time goes to infinity, the solution converges in Cloc2 (Rn) to a solution to the corresponding elliptic problem. In particular, we show that for any two solutions u1(x′) 0, in Rn such that lim xn→+∞ u(x′, xn) = v1(x′) < u1(x′) and limxn→+∞ u(x′, xn) = v2(x′) 1. Some of our results are similar to results for minimizers obtained by Jerison and Monneau [13] by variational arguments. The novelty of our paper is that we only assume the condition for F used by Keller and Osserman for boundary blow up solutions.