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The concentration compactness principle of P.-L.Lions can be formulated in terms of general Hilbert space. Further applications to Sobolev spaces are given.

Let a be a quadratic form associated with a Schrödinger operator L = −∇ · (A∇) + V on a domain Ω ⊂ R d. 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 ϕ k ∈ C ∞ 0 (Ω) and a function ϕ > 0 satisfying Lϕ = 0 such that a[ϕ k ] → 0, ϕ k → ϕ locally uniformly in Ω \ {x 0 }.… (More)

- K. Tintarev
- 2008

We show existence of minimizers for the Hardy-Sobolev-Maz'ya inequality in R m+n \ R n when either m > 2, n ≥ 1 or m = 1, n ≥ 3.

Let Ω be a domain in R d , d ≥ 2, and 1 < p < ∞. Fix V ∈ L ∞ loc (Ω). Consider the functional Q and its Gâteaux derivative Q ′ given by Q(u) := Ω (|∇u| p +V |u| p)dx, 1 p Q ′ (u) := −∇·(|∇u| p−2 ∇u)+V |u| p−2 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… (More)

- Yehuda Pinchover, Achilles Tertikas, Kyril Tintarev
- 2008

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.

The paper studies the existence of minimizers for Rayleigh quotients µ Ω = 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 .

- I Schindler, K Tintarev
- 2001

The paper presents an existence result for a nonlinear Schrödinger equation with magnetic potential on unbounded domains.

We present the most general definition of the linking of sets in a Hilbert space and, drawing on the theory given in [ST, T, Sc3], give a necessary and sufficient geometric condition for linking when one set is compact.

- Kyril Tintarev
- 2009

The paper raises a question about the optimal critical nonlinearity for the Sobolev space in two dimensions, connected to loss of compact-ness, and discusses the pertinent concentration compactness framework. We study properties of the improved version of the Trudinger-Moser inequality on the open unit disk B ⊂ R 2 , recently proved by G. Mancini and K.… (More)