Abstract
AbstractConsider the boundary blow-up Monge-Ampère problem$$\begin{array}{} \displaystyle M[u]=K(x)f(u) \mbox{ for } x \in {\it\Omega},\; u(x)\rightarrow +\infty \mbox{ as } {\rm dist}(x,\partial {\it\Omega})\rightarrow 0. \end{array}$$HereM[u] = det (uxixj) is the Monge-Ampère operator, andΩis a smooth, bounded, strictly convex domain in ℝN(N≥ 2). UnderK(x) satisfying appropriate conditions, we first prove that the boundary blow-up Monge-Ampère problem has a strictly convex solution if and only iffsatisfies Keller-Osserman type condition. Then the asymptotic behavior of strictly convex solutions to the boundary blow-up Monge-Ampère problem is considered under weaker condition with respect to previous references. Finally, iffdoes not satisfy Keller-Osserman type condition, we show the existence of strictly convex solutions under different conditions onK(x). The proof combines standard techniques based upon the sub-supersolution method with non-standard arguments, such as the Karamata regular variation theory.
Highlights
Monge-Ampère problems are fully nonlinear problems, which can describe Weingarten curvature, or re ector shape design
Under K(x) satisfying appropriate conditions, we rst prove that the boundary blow-up MongeAmpère problem has a strictly convex solution if and only if f satis es Keller-Osserman type condition
The asymptotic behavior of strictly convex solutions to the boundary blow-up Monge-Ampère problem is considered under weaker condition with respect to previous references
Summary
Monge-Ampère problems are fully nonlinear problems, which can describe Weingarten curvature, or re ector shape design (see [1]). The asymptotic behavior of strictly convex solutions to the boundary blow-up Monge-Ampère problem is considered under weaker condition with respect to previous references. Where M[u] = det (uxixj ) is the Monge-Ampère operator, Ω is a smooth, bounded, strictly convex domain in RN(N ≥ ), and K(x), f (u) are smooth positive functions.
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