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15
A threshold phenomenon for embeddings of H m 0 into Orlicz spaces
, 2009
"... Given an open bounded domain Ω ⊂ R 2m with smooth boundary, we consider a sequence (uk)k∈N of positive smooth solutions to j m (−∆) uk = λkuke mu2 k in Ω uk = ∂νuk =... = ∂ m−1 ν uk = 0 on ∂Ω, where λk → 0 +. Assuming that the sequence is bounded in H m 0 (Ω), we study its blowup behavior. We show ..."
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Given an open bounded domain Ω ⊂ R 2m with smooth boundary, we consider a sequence (uk)k∈N of positive smooth solutions to j m (−∆) uk = λkuke mu2 k in Ω uk = ∂νuk =... = ∂ m−1 ν uk = 0 on ∂Ω, where λk → 0 +. Assuming that the sequence is bounded in H m 0 (Ω), we study its blowup behavior. We show that if the sequence is not precompact, then lim inf k→ ∞ ‖uk ‖ 2 H m 0 Z: = lim inf uk(−∆) k→∞
Conformal metrics on R2m with constant Qcurvature and large volume
, 2012
"... We study conformal metrics gu = e 2udx2 on R2m with constant Qcurvature Qgu ≡ (2m − 1)! (notice that (2m − 1)! is the Qcurvature of S2m) and finite volume. When m = 3 we show that there exists V ∗ such that for any V ∈ [V ∗,∞) there is a conformal metric gu = e 2udx2 on R6 with Qgu ≡ 5! and vo ..."
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We study conformal metrics gu = e 2udx2 on R2m with constant Qcurvature Qgu ≡ (2m − 1)! (notice that (2m − 1)! is the Qcurvature of S2m) and finite volume. When m = 3 we show that there exists V ∗ such that for any V ∈ [V ∗,∞) there is a conformal metric gu = e 2udx2 on R6 with Qgu ≡ 5! and vol(gu) = V. This is in sharp contrast with the fourdimensional case, treated by CS. Lin. We also prove that when m is odd and greater than 1, there is a constant Vm> vol(S 2m) such that for every V ∈ (0, Vm] there is a conformal metric gu = e 2udx2 on R2m with Qgu ≡ (2m − 1)!, vol(g) = V. This extends a result of A. Chang and WX. Chen. When m is even we prove a similar result for conformal metrics of negative Qcurvature.
Conformal metrics on R 2m with constant Qcurvature
, 2008
"... We study the conformal metrics on R 2m with constant Qcurvature Q ∈ R having finite volume, particularly in the case Q ≤ 0. We show that when Q < 0 such metrics exist in R 2m if and only if m> 1. Moreover we study their asymptotic behavior at infinity, in analogy with the case Q> 0, which ..."
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We study the conformal metrics on R 2m with constant Qcurvature Q ∈ R having finite volume, particularly in the case Q ≤ 0. We show that when Q < 0 such metrics exist in R 2m if and only if m> 1. Moreover we study their asymptotic behavior at infinity, in analogy with the case Q> 0, which we treated in a recent paper. When Q = 0, we show that such metrics have the form e 2p g R 2m, where p is a polynomial such that 2 ≤ deg p ≤ 2m − 2 and sup R 2m p < +∞. In dimension 4, such metrics are exactly the polynomials p of degree 2 with limx→+ ∞ p(x) = −∞. 1 Introduction and statement of the main theorems Given a constant Q ∈ R, we consider the solutions to the equation (−∆) m u = Qe 2mu on R 2m, (1) satisfying
COMPACTNESS ISSUES AND BUBBLING PHENOMENA FOR THE PRESCRIBED GAUSSIAN CURVATURE EQUATION ON THE TORUS
, 2014
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unknown title
, 2009
"... Asymptotics and quantization for a meanfield equation of higher order ..."
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where Ω0
, 2010
"... We discuss compactness, blowup and quantization phenomena for the prescribed Qcurvature equation (−∆) m uk = Vke 2muk on open domains of R 2m. Under natural integral assumptions we show that when blowup occurs, up to a subsequence lim Vke ..."
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We discuss compactness, blowup and quantization phenomena for the prescribed Qcurvature equation (−∆) m uk = Vke 2muk on open domains of R 2m. Under natural integral assumptions we show that when blowup occurs, up to a subsequence lim Vke