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Keywords:
Sobolev space; Orlicz-Sobolev space; Moser-Trudinger inequality; sharp constant; concentration-compactness principle
Sobolev space; Orlicz-Sobolev space; Moser-Trudinger inequality; sharp constant; concentration-compactness principle
Summary:
Let $\Omega \subset \mathbb R^n$ be a domain and let $\alpha <n-1$. We prove the Concentration-Compactness Principle for the embedding of the space $W_0^1L^n\log ^{\alpha }L(\Omega )$ into an Orlicz space corresponding to a Young function which behaves like $\exp (t^{{n}/{(n-1-\alpha )}})$ for large $t$. We also give the result for the embedding into multiple exponential spaces. \endgraf Our main result is Theorem \ref {lions4} where we show that if one passes to unbounded domains, then, after the usual modification of the integrand in the Moser functional, the statement of the Concentration-Compactnes Principle is very similar to the statement in the case of a bounded domain. In particular, in the case of a nontrivial weak limit the borderline exponent is still given by the formula $$ P:=(1-\|\Phi (|\nabla u|)\|_{L^1(\mathbb R^n)})^{-{1}/{(n-1)}}. $$
References:
[1] Adachi, S., Tanaka, K.: Trudinger type inequalities in $\mathbb R^N$ and their best exponents. Proc. Am. Math. Soc. 128 2051-2057 (2000). DOI 10.1090/S0002-9939-99-05180-1 | MR 1646323
[2] Adimurthi: Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the $n$-Laplacian. Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 17 393-413 (1990). MR 1079983 | Zbl 0732.35028
[3] Battaglia, L., Mancini, G.: Remarks on the Moser-Trudinger inequality. Adv. Nonlinear Anal. 2 389-425 (2013). MR 3199739 | Zbl 1290.46025
[4] Carleson, L., Chang, S.-Y. A.: On the existence of an extremal function for an inequality of J. Moser. Bull. Sci. Math., II. Sér. 110 113-127 (1986), French summary. MR 0878016
[5] Černý, R.: Concentration-compactness principle for embedding into multiple exponential spaces. Math. Inequal. Appl. 15 165-198 (2012). MR 2919441 | Zbl 1236.46027
[6] Černý, R.: Generalized Moser-Trudinger inequality for unbounded domains and its application. NoDEA, Nonlinear Differ. Equ. Appl. 19 575-608 (2012). DOI 10.1007/s00030-011-0143-0 | MR 2984597 | Zbl 1262.46025
[7] Černý, R.: Note on the Concentration-compactness principle for generalized Moser-Trudinger inequalities. Cent. Eur. J. Math. 10 590-602 (2012). DOI 10.2478/s11533-011-0102-3 | MR 2893423 | Zbl 1272.46019
[8] Černý, R., Cianchi, A., Hencl, S.: Concentration-compactness principle for Moser-Trudinger inequalities: new results and proofs. Ann. Mat. Pura Appl. (4) 192 225-243 (2013). DOI 10.1007/s10231-011-0220-3 | MR 3035137
[9] Černý, R., Gurka, P., Hencl, S.: Concentration-compactness principle for generalized Trudinger inequalities. Z. Anal. Anwend. 30 355-375 (2011). DOI 10.4171/ZAA/1439 | MR 2819500 | Zbl 1225.46026
[10] Černý, R., Gurka, P., Hencl, S.: On the Dirichlet problem for the $n,\alpha$-Laplacian with the nonlinearity in the critical growth range. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74 5189-5204 (2011). DOI 10.1016/j.na.2011.05.015 | MR 2810699 | Zbl 1225.35062
[11] Černý, R., Mašková, S.: A sharp form of an embedding into multiple exponential spaces. Czech. Math. J. 60 751-782 (2010). DOI 10.1007/s10587-010-0048-9 | MR 2672414 | Zbl 1224.46064
[12] Chabrowski, J.: Concentration-compactness principle at infinity and semilinear elliptic equations involving critical and subcritical Sobolev exponents. Calc. Var. Partial Differ. Equ. 3 493-512 (1995). DOI 10.1007/BF01187898 | MR 1385297 | Zbl 0838.35035
[13] Cianchi, A.: A sharp embedding theorem for Orlicz-Sobolev spaces. Indiana Univ. Math. J. 45 39-65 (1996). DOI 10.1512/iumj.1996.45.1958 | MR 1406683 | Zbl 0860.46022
[14] Figueiredo, D. G. de, Miyagaki, O. H., Ruf, B.: Elliptic equations in $\mathbb R^2$ with nonlinearities in the critical growt hrange. Calc. Var. Partial Differ. Equ. 3 139-153 (1995). DOI 10.1007/BF01205003 | MR 1386960
[15] 'O, J. M. do: $N$-Laplacian equations in $\mathbb R^N$ with critical growth. Abstr. Appl. Anal. 2 301-315 (1997). DOI 10.1155/S1085337597000419 | MR 1704875
[16] Ó, J. M. do, Souza, M. de, Medeiros, E. de, Severo, U.: An improvement for the Trudinger-Moser inequality and applications. J. Differ. Equations 256 1317-1349 (2014). DOI 10.1016/j.jde.2013.10.016 | MR 3145759
[17] Ó, J. M. do, Medeiros, E., Severo, U.: On a quasilinear nonhomogenous elliptic equation with critical growth in $\mathbb R^N$. J. Differ. Equations 246 1363-1386 (2009). DOI 10.1016/j.jde.2008.11.020 | MR 2488689
[18] Edmunds, D. E., Gurka, P., Opic, B.: Norms of embeddings of logarithmic Bessel potential spaces. Proc. Am. Math. Soc. 126 2417-2425 (1998). DOI 10.1090/S0002-9939-98-04327-5 | MR 1451796 | Zbl 0895.46020
[19] Edmunds, D. E., Gurka, P., Opic, B.: On embeddings of logarithmic Bessel potential spaces. J. Funct. Anal. 146 116-150 (1997). DOI 10.1006/jfan.1996.3037 | MR 1446377 | Zbl 0934.46036
[20] Edmunds, D. E., Gurka, P., Opic, B.: Sharpness of embeddings in logarithmic Bessel-potential spaces. Proc. R. Soc. Edinb., Sect. A 126 995-1009 (1996). DOI 10.1017/S0308210500023210 | MR 1415818 | Zbl 0860.46024
[21] Edmunds, D. E., Gurka, P., Opic, B.: Double exponential integrability, Bessel potentials and embedding theorems. Stud. Math. 115 151-181 (1995). MR 1347439 | Zbl 0829.47024
[22] Edmunds, D. E., Gurka, P., Opic, B.: Double exponential integrability of convolution operators in generalized Lorentz-Zygmund spaces. Indiana Univ. Math. J. 44 19-43 (1995). DOI 10.1512/iumj.1995.44.1977 | MR 1336431 | Zbl 0826.47021
[23] Edmunds, D. E., Krbec, M.: Two limiting cases of Sobolev imbeddings. Houston J. Math. 21 119-128 (1995). MR 1331250 | Zbl 0835.46027
[24] Fusco, N., Lions, P.-L., Sbordone, C.: Sobolev imbedding theorems in borderline cases. Proc. Am. Math. Soc. 124 561-565 (1996). DOI 10.1090/S0002-9939-96-03136-X | MR 1301025 | Zbl 0841.46023
[25] Hencl, S.: A sharp form of an embedding into exponential and double exponential spaces. J. Funct. Anal. 204 (2003), 196-227. DOI 10.1016/S0022-1236(02)00172-6 | MR 2004749 | Zbl 1034.46031
[26] Li, Y., Ruf, B.: A sharp Trudinger-Moser type inequality for unbounded domains in $\mathbb R^n$. Indiana Univ. Math. J. 57 451-480 (2008). DOI 10.1512/iumj.2008.57.3137 | MR 2400264
[27] Lions, P.-L.: The concentration-compactness principle in the calculus of variations. The limit case. I. Rev. Mat. Iberoam. 1 (1985), 145-201. DOI 10.4171/RMI/6 | MR 0834360 | Zbl 0704.49005
[28] Moser, J.: A sharp form of an inequality by Trudinger. Indiana Univ. Math. J. 20 1077-1092 (1971). DOI 10.1512/iumj.1971.20.20101 | MR 0301504 | Zbl 0213.13001
[29] Opic, B., Pick, L.: On generalized Lorentz-Zygmund spaces. Math. Inequal. Appl. 2 391-467 (1999). MR 1698383 | Zbl 0956.46020
[30] Rao, M. M., Ren, Z. D.: Theory of Orlicz Spaces. Pure and Applied Mathematics 146 Marcel Dekker, New York (1991). MR 1113700 | Zbl 0724.46032
[31] Ruf, B.: A sharp Trudinger-Moser type inequality for unbounded domains in $\mathbb R^2$. J. Funct. Anal. 219 340-367 (2005). DOI 10.1016/j.jfa.2004.06.013 | MR 2109256
[32] Talenti, G.: Inequalities in rearrangement invariant function spaces. Nonlinear Analysis, Function Spaces and Applications. Vol. 5 M. Krbec et al. Proc. Conf., Praha, 1994. Prometheus Publishing House Praha (1994), 177-230. MR 1322313 | Zbl 0872.46020
[33] Trudinger, N. S.: On imbeddings into Orlicz spaces and some applications. J. Math. Mech. 17 473-484 (1967). MR 0216286 | Zbl 0163.36402
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