Asymptotic boundary estimates to infinity Laplace equations with Γ-varying nonlinearity

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:436

Article

Wang, Wei¹  Gong, Hanzhao¹  He, Xiao¹  Zheng, Sining¹ 

[1] Dalian Univ Technol, Sch Math Sci, Dalian 116024, Peoples R China

关键词: Infinity Laplacian;    Asymptotic estimates;    Boundary blow-up;    Gamma-varying;    Comparison principle;   

DOI  :  10.1016/j.jmaa.2015.11.046

来源: Elsevier

【 摘 要 】

In a previous paper of the authors (Wang et al. (2014) [40]), the asymptotic estimates of boundary blow-up solutions were established to the infinity Laplace equation Delta infinity u = b(x)f (u) in Omega subset of R-N, with the nonlinearity 0 <= f is an element of C[0, infinity) regularly varying at infinity, and the weighted function b is an element of C((Omega) over bar) positive in Omega and vanishing on the boundary. The present paper gives a further investigation on the asymptotic behavior of boundary blow-up solutions to the same equation by assuming f to be Gamma-varying. Note that a Gamma-varying function grows faster than any regularly varying function. We first quantitatively determine the boundary blow-up estimates with the first expansion, relying on the decay rate of b near the boundary and the growth rate of f at infinity, and further characterize these results via examples possessing various decay rates for b and growth rates for f. In particular, we pay attention to the second-order estimates of boundary blow-up solutions. It was observed in our previous work that the second expansion of solutions to the infinity Laplace equation is independent of the geometry of the domain, quite different from the classical Laplacian. The second expansion obtained in this paper furthermore shows a substantial difference on the asymptotic behavior of boundary blow-up solutions between the infinity Laplacian and the classical Laplacian. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

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