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On attainability of Moser-Trudinger inequality with logarithmic weights in higher dimensions

Part of this research is supported by Inspire programme under the contract number IFA14/MA-43

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  • Moser-Trudinger inequality was generalised by Calanchi-Ruf to the following version: If $ \beta \in [0,1) $ and $ w_0(x) = |\log |x||^{\beta(n-1)} $ or $ \left( \log \frac{e}{|x|}\right)^{\beta(n-1)} $ then

    $ \sup\limits_{\int_B | \nabla u|^nw_0 \leq 1 , u \in W_{0,rad}^{1,n}(w_0,B)} \;\; \int_B \exp\left(\alpha |u|^{\frac{n}{(n-1)(1-\beta)}} \;\;\; \right) dx < \infty $

    if and only if $ \alpha \leq \alpha_\beta = n\left[\omega_{n-1}^{\frac{1}{n-1}}(1-\beta) \right]^{\frac{1}{1-\beta}} $ where $ \omega_{n-1} $ denotes the surface measure of the unit sphere in $ \mathbb {R}^n $. The primary goal of this work is to address the issue of existence of extremal function for the above inequality. A non-existence (of extremal function) type result is also discussed, for the usual Moser-Trudinger functional.

    Mathematics Subject Classification: Primary: 35B38, 35J20, 47N20, 26D10; Secondary: 46E35.

    Citation:

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  • [1] Adimurthi and K. Sandeep, A singular Moser-Trudinger embedding and its applications, NoDEA Nonlinear Differential Equations Appl., 13 (2007), 585-603.  doi: 10.1007/s00030-006-4025-9.
    [2] Adimurthi and C. Tintarev, On compactness in the Trudinger-Moser inequality, Ann. Sc. Norm. Super. Pisa Cl. Sci., 13 (2014), 399-416. 
    [3] M. Calanchi, Some weighted inequalities of Trudinger–Moser type, Progress in Nonlinear Differential Equations and Appl., 85 (2014), 163-174. 
    [4] M. Calanchi and B. Ruf, Trudinger-Moser type inequalities with logarithmic weights in dimension $N$, Nonlinear Anal., 121 (2015), 403-411.  doi: 10.1016/j.na.2015.02.001.
    [5] M. Calanchi and B. Ruf, On Trudinger-Moser type inequalities with logarithmic weights, J. Differential Equations, 258 (2015), 1967-1989.  doi: 10.1016/j.jde.2014.11.019.
    [6] L. Carleson and S.-Y. A. Chang, On the existence of an extremal function for an inequality by J. Moser, Bull. Sci. Math., 110 (1986), 113-127. 
    [7] G. Csató and P. Roy, Extremal functions for the singular Moser-Trudinger inequality in 2 dimensions, Calc. Var. Partial Differential Equations, 54 (2015), 2341-2366.  doi: 10.1007/s00526-015-0867-5.
    [8] G. Csató and P. Roy, The singular Moser-Trudinger inequality on simply connected domain, Comm. Partial Differential Equations, 41 (2016), 838-847.  doi: 10.1080/03605302.2015.1123276.
    [9] G. Csató, N. H. Nguyen and P. Roy, Extremals for the Singular Moser-Trudinger Inequality via n-Harmonic Transplantation, preprint, arXiv: 1801.03932v3.
    [10] D. G. de FigueiredoJ. M. do O and B. Ruf, Elliptic equations and systems with critical Trudinger-Moser nonlinearities., Discrete Contin. Dyn. Syst., 30 (2011), 455-476.  doi: 10.3934/dcds.2011.30.455.
    [11] M. Flucher, Extremal functions for the Trudinger-Moser inequality in 2 dimensions, Comment. Math. Helvetici, 67 (1992), 471-497.  doi: 10.1007/BF02566514.
    [12] N. Lam, Equivalence of sharp Trudinger-Moser-Adams inequalities, Commun. Pure Appl. Anal, 16 (2017), 973-997.  doi: 10.3934/cpaa.2017047.
    [13] N. Lam and G. Lu, A new approach to sharp Moser-Trudinger and Adams type inequalities: A rearrangement-free argument., J. Differential Equations, 255 (2013), 298-325.  doi: 10.1016/j.jde.2013.04.005.
    [14] X. Li and Y. Yang, Extremal functions for singular Trudinger-Moser inequalities in the entire Euclidean space, prerpint, arXiv: 1612.08247
    [15] K.-C. Lin, Extremal functions for Moser's inequality, Trans. Amer. Math. Soc., 348 (1996), 2663-2671.  doi: 10.1090/S0002-9947-96-01541-3.
    [16] P.-L. Lions, The concentration-compactness principle in the Calculus of variations. The limit case. Ⅰ, Rev. Mat. Iberoamericana, 1 (1985), 145-201.  doi: 10.4171/RMI/6.
    [17] G. Lu and H. Tang, Best constants for Moser-Trudinger inequalities on high dimensional hyperbolic spaces, Adv. Nonlinear Stud., 13 (2013), 1035-1052.  doi: 10.1515/ans-2013-0415.
    [18] G. Lu and Y. Yang, Sharp constant and extremal function for the improved Moser-Trudinger inequality involving $L^p$ norm in two dimension, Discrete Contin. Dyn. Syst., 25 (2009), 963-979.  doi: 10.3934/dcds.2009.25.963.
    [19] S. Lula and G. Mancini, Extremal functions for singular Moser-Trudinger embeddings, Nonlinear Anal., 156 (2017), 215-248.  doi: 10.1016/j.na.2017.02.029.
    [20] A. Malchiodi and L. Martinazzi, Critical points of the Moser-Trudinger functional on a disk, J. Eur. Math. Soc. (JEMS), 16 (2014), 893-908.  doi: 10.4171/JEMS/450.
    [21] G. Mancini and L. Battaglia, Remarks on the Moser-Trudinger inequality, Adv. Nonlinear Anal, 2 (2013), 389-425.  doi: 10.1515/anona-2013-0014.
    [22] G. Mancini and K. Sandeep, Moser-Trudinger inequality on conformal discs, Commun. Contemp. Math, 12 (2010), 1055-1068.  doi: 10.1142/S0219199710004111.
    [23] G. ManciniK. Sandeep and C. Tintarev, Trudinger-Moser inequality in the hyperbolic space $H^N$, Adv. Nonlinear Anal, 2 (2013), 309-324.  doi: 10.1515/anona-2013-0001.
    [24] J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J, 20 (1971), 1077-1092.  doi: 10.1512/iumj.1971.20.20101.
    [25] Q.-A. Ngo and V. H. Nguyen, An improved Moser-Trudinger inequality involving the first non-zero Neumann eigenvalue with mean value zero in $\mathbb{R}^2$, preprint, arXiv: 1702.08883
    [26] V. H. Nguyen, A sharp Adams inequality in dimension four and its extremal functions, arXiv: 1701.08249
    [27] V. H. Nguyen and F. Takahashi, On a weighted Trudinger-Moser type inequality on the whole space and its (non-)existence of maximizers, Differential Integral Equations, 31 (2018), 785-806. 
    [28] P. Roy, Extremal function for Moser-Trudinger type inequality with logarithmic weight, Nonlinear Anal., 135 (2016), 194-204.  doi: 10.1016/j.na.2016.01.024.
    [29] B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $R^2$, J. Funct. Anal, 219 (2005), 340-367.  doi: 10.1016/j.jfa.2004.06.013.
    [30] M. Struwe, Critical points of embeddings of $H_0^{1, n}$ into Orlicz spaces, Ann. Inst. H. Poincaré Anal. Non Linéaire, 5 (1988), 425-464.  doi: 10.1016/S0294-1449(16)30338-9.
    [31] N. S. Trudinger, On embeddings into Orlicz spaces and some applications, J. Math. Mech, 17 (1967), 473-484.  doi: 10.1512/iumj.1968.17.17028.
    [32] Y. Yang, Extremal functions for a sharp Moser-Trudinger inequality, Internat. J. Math., 17 (2006), 331-338.  doi: 10.1142/S0129167X06003503.
    [33] Y. Yang, Extremal functions for Moser-Trudinger inequalities on 2-dimensional compact Riemannian manifolds with boundary, Internat. J. Math., 17 (2006), 313-330.  doi: 10.1142/S0129167X06003473.
    [34] Y. Yang, Extremal functions for Trudinger-Moser inequalities of Adimurthi-Druet type in dimension two, J. Differential Equations, 258 (2015), 3161-3193.  doi: 10.1016/j.jde.2015.01.004.
    [35] Y. Yang, A Trudinger-Moser inequality on a compact Riemannian surface involving Gaussian curvature, J. Geom. Anal., 26 (2016), 2893-2913.  doi: 10.1007/s12220-015-9653-z.
    [36] X. Zhu and Y. Yang, Blow-up analysis concerning singular Trudinger Moser inequalities in dimension two, J. Funct. Anal, 272 (2017), 3347-3374.  doi: 10.1016/j.jfa.2016.12.028.
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