Kyril Tintarev

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The paper studies the existence of minimizers for Rayleigh quotients μΩ = inf ∫ Ω |∇u| 2 ∫ Ω V |u| 2 , where Ω is a domain in R N , and V is a nonzero nonnegative function that may have singularities on ∂Ω. As a model for our results one can take Ω to be a Lipschitz cone and V to be the Hardy potential V (x) = 1 |x|2 . 2000 Mathematics Subject(More)
Let a be a quadratic form associated with a Schrödinger operator L = −∇ · (A∇) + V on a domain Ω ⊂ R. If a is nonnegative on C∞ 0 (Ω), then either there is W > 0 such that ∫ W |u|2 dx ≤ a[u] for all C∞ 0 (Ω;R), or there is a sequence φk ∈ C ∞ 0 (Ω) and a function φ > 0 satisfying Lφ = 0 such that a[φk] → 0, φk → φ locally uniformly in Ω \ {x0}. This(More)
Let Ω be a domain in Rd, d ≥ 2, and 1 < p <∞. Fix V ∈ Lloc(Ω). Consider the functional Q and its Gâteaux derivative Q′ given by Q(u) := ∫ Ω (|∇u|+V |u|)dx, 1 p Q(u) := −∇·(|∇u|∇u)+V |u|u. If Q ≥ 0 on C∞ 0 (Ω), then either there is a positive continuous function W such that ∫ W |u|p dx ≤ Q(u) for all u ∈ C∞ 0 (Ω), or there is a sequence uk ∈ C ∞ 0 (Ω) and a(More)
In this paper we prove a sufficient condition, in terms of the behavior of a ground state of a singular p-Laplacian problem with a potential term, such that a nonzero subsolution of another such problem is also a ground state. Unlike in the linear case (p = 2), this condition involves comparison of both the functions and of their gradients. 2000 Mathematics(More)
Let Ω be a domain in R (possibly unbounded), N ≥ 2, 1 < p <∞, and let V ∈ Lloc(Ω). Consider the energy functional QV on C∞ c (Ω) and its Gâteaux derivative QV , respectively, given by QV (u) def = 1 p ∫ Ω (|∇u| + V |u|) dx, QV (u) = div(|∇u|p−2∇u) + V |u|p−2u, for u ∈ C∞ c (Ω). Assume that QV > 0 on C∞ c (Ω) \ {0} and QV does not have a ground state (in the(More)