Journal ArticleDOI
Qualitative analysis of singular solutions for nonlinear elliptic equations with potentials
Jann Long Chern,Eiji Yanagida +1 more
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In this paper, the structure of radially symmetric singular solutions for elliptic equations with the Hardy term and power nonlinearity was studied. But the authors only considered the subcritical and supercritical cases and made clear the difference of structure from the critical case.Abstract:
We consider the structure of radially symmetric singular solutions for elliptic equations with the Hardy term and power nonlinearity. In the critical case, it is shown that there exists a unique non-oscillatory singular solution, around which infinitely many singular solutions are oscillating. We also study the subcritical and supercritical cases and make clear the difference of structure from the critical case. Our results can be applied to various problems such as the minimizing problem related to the Caffarelli–Kohn–Nirenberg inequality, the scalar field equation and a self-replication model.read more
Citations
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Intersection properties for singular radial solutions of quasilinear elliptic equations with Hardy type potentials
TL;DR: In this article, singular positive solutions of a quasilinear elliptic equation with singular coefficient r−(γ−1), rα|u′|β−1u′)′+krα−β−γuβ+up=0,0 β, 0
Liouville Theorem for semilinear elliptic inequalities with the fractional Hardy operators
TL;DR: In this paper , the authors give a full classification of the nonexistence of positive weak solutions to the semilinear elliptic inequality involving the fractional Hardy potential in punctured and in exterior domains.
References
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Journal ArticleDOI
Positive solutions of nonlinear elliptic equations involving critical sobolev exponents
Haim Brezis,Louis Nirenberg +1 more
TL;DR: In this article, the existence of a fonction u satisfaisant l'equation elliptique non lineaire is investigated, i.e., a domaine borne in R n avec n ≥ 3.
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Nonlinear scalar field equations, I existence of a ground state
TL;DR: In this article, a constrained minimization method was proposed for the case of dimension N = 1 (Necessary and sufficient conditions) for the zero mass case, where N is the number of dimensions in the dimension N.
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Best constant in Sobolev inequality
TL;DR: The best constant for the simplest Sobolev inequality was proved in this paper by symmetrizations (rearrangements in the sense of Hardy-Littlewood) and one-dimensional calculus of variations.
Journal ArticleDOI
Uniqueness of positive solutions of Δu−u+up=0 in Rn
TL;DR: In this article, the uniqueness of the positive, radially symmetric solution to the differential equation Δu−u+up=0 (with p>1) in a bounded or unbounded annular region in Rn for all n ≥ 1, with the Neumann boundary condition on the inner ball and the Dirichlet boundary condition decaying to zero in the case of an unbounded region, was established.
Book ChapterDOI
Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities
TL;DR: For the doubly weighted HLS inequality, the authors showed that f and N are explicitly evaluated when p =q' or p = 2 or q = 2, when q' or q' < 2, and when n = 2n/(2b + n - 2).
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