Boundary behavior of large solutions to the Monge–Ampère equations with weights

Abstract

This paper is concerned with the Monge-Ampère equation detwhere is a strictly convex, bounded smooth domain in R N with N ≥ 2, and b ∈ C ∞ (¯ ) which is positive in , but may be vanishing on the boundary. We find a new structure condition on f which plays a crucial role in the boundary behavior of strictly convex large solutions. Our results are obtained in a more general setting than those in Cîrstea and Trombetti (2008) [12], where f is regularly varying at infinity with index p > N.

Existence and boundary asymptotic behavior of large solutions of Hessian equations

“…In fact, we generalize the asymptotic results for Monge-Ampère equation in [28] to k-Hessian equation. Our results are also more accurate than [14].…”

“…For detailed proofs of Lemmas 3.1 and 3.2, we refer to Lemmas 2.1 and 2.3 of [28] respectively. More characterization of functions in (f 2 ) and (b 2 ) are also provided there.…”

Existence of entire positivek-convex radial solutions to Hessian equations and systems with weights

“…The results have been extended by many authors and in many contexts, see, for instance, [19][20][21][22][23][24][25] and the references therein. Now let us return to (1.1).…”

Asymptotic behavior of boundary blow-up solutions to elliptic equations

“…However, the case that (2.1) (b) and (2.2) (b) hold as well as the case that j = −1 in (2.1) (a) is coupled with υ = −1 in (2.2) (a) were left open in [11]. For the other related results on boundary blow-up solutions to (1.1) and related problems, we refer to [16,20,27,31,32,39,40,43,46,47] and the references therein.…”