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On the large solutions to a class of k-Hessian problems
- Source :
- Advanced Nonlinear Studies, Vol 24, Iss 3, Pp 657-695 (2024)
- Publication Year :
- 2024
- Publisher :
- De Gruyter, 2024.
-
Abstract
- In this paper, we consider the k-Hessian problem S k(D 2 u) = b(x)f(u) in Ω, u = +∞ on ∂Ω, where Ω is a C ∞-smooth bounded strictly (k − 1)-convex domain in RN ${\mathbb{R}}^{N}$ with N ≥ 2, b ∈ C∞(Ω) is positive in Ω and may be singular or vanish on ∂Ω, f ∈ C[0, ∞) ∩ C 1(0, ∞) (or f∈C1(R) $f\in {C}^{1}\left(\mathbb{R}\right)$ ) is a positive and increasing function. We establish the first expansions (equalities) of k-convex solutions to the above problem when f is borderline regularly varying and Γ-varying at infinity respectively. For the former, we reveal the exact influences of some indexes of f and principal curvatures of ∂Ω on the first expansion of solutions. For the latter, we find the principal curvatures of ∂Ω have no influences on the expansions. Our results and methods are quite different from the existing ones (including k = N). Moreover, we know the existence of k-convex solutions to the above problem (including k = N) is still an open problem when b possesses high singularity on ∂Ω and f satisfies Keller–Osserman type condition. For the radially symmetric case in the ball, we give a positive answer to this open problem, and then we further show the global estimates for all radial large solutions.
Details
- Language :
- English
- ISSN :
- 21690375
- Volume :
- 24
- Issue :
- 3
- Database :
- Directory of Open Access Journals
- Journal :
- Advanced Nonlinear Studies
- Publication Type :
- Academic Journal
- Accession number :
- edsdoj.1dc660a3206f4723a0e04e9cd8ce18bb
- Document Type :
- article
- Full Text :
- https://doi.org/10.1515/ans-2023-0128