2011, 30(2): 455-476. doi: 10.3934/dcds.2011.30.455

Elliptic equations and systems with critical Trudinger-Moser nonlinearities

1. 

IMECC-UNICAMP, Caixa Postal 6065, CEP: 13081-970, Campinas - SP, Brazil

2. 

Departamento de Matemática–Universidade Federal da Paraíba, 58051-900, João Pessoa–PB

3. 

Dip. di Matematica, Universita degli Studi, Via Saldini 50, 20133 Milano, Italy

Received  April 2010 Published  February 2011

In this article we give first a survey on recent results on some Trudinger-Moser type inequalities, and their importance in the study of nonlinear elliptic equations with nonlinearities which have critical growth in the sense of Trudinger-Moser. Furthermore, recent results concerning systems of such equations will be discussed.
Citation: Djairo G. De Figueiredo, João Marcos do Ó, Bernhard Ruf. Elliptic equations and systems with critical Trudinger-Moser nonlinearities. Discrete & Continuous Dynamical Systems - A, 2011, 30 (2) : 455-476. doi: 10.3934/dcds.2011.30.455
References:
[1]

S. Adachi and K. Tanaka, Trudinger type inequalities in $\mathbbR^N$ and their best exponents,, Proc. Amer. Math. Soc., 128 (2000), 2051. doi: 10.1090/S0002-9939-99-05180-1.

[2]

D. R. Adams, A sharp inequality of J. Moser for higher order derivatives,, Annals of Math., 128 (1988), 385. doi: 10.2307/1971445.

[3]

R. A. Adams and J. F. Fournier, "Sobolev Spaces,'', Second Edition, (2003).

[4]

Adimurthi, Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the $n$-Laplacian,, Ann. Sc. Norm. Sup. Pisa, XVII (1990), 393.

[5]

Adimurthi and O. Druet, Blow-up analysis in dimension 2 and a sharp form of Trudinger-Moser inequality,, Comm. Part. Diff. Equ., 29 (2004), 295. doi: 10.1081/PDE-120028854.

[6]

Adimurthi and K. Sandeep, A singular Moser-Trudinger embedding and its applications,, NoDEA Nonlinear Differential Equations Appl., 13 (2007), 585. doi: 10.1007/s00030-006-4025-9.

[7]

A. Alvino, V. Ferone and G. Trombetti, Moser-type inequalities in Lorentz spaces,, Potential Anal., 5 (1996), 273.

[8]

A. Alvino, P.-L. Lions and G. Trombetti, On optimization problems with prescribed rearrangements,, Nonlinear Anal. T.M.A., 13 (1989), 185. doi: 10.1016/0362-546X(89)90043-6.

[9]

V. V. Atkinson and L. A. Peletier, Ground states and Dirichlet problems for $-\Delta u = f(u)$ in $\mathbb R^2$,, Arch. Rational Mech. Anal., 96 (1986), 147. doi: 10.1007/BF00251409.

[10]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications,, J. Functional Analysis, 14 (1973), 349. doi: 10.1016/0022-1236(73)90051-7.

[11]

A. Bahri and J.-M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: The effect of topology of the domain,, Comm. Pure Appl. Math., 41 (1988), 253. doi: 10.1002/cpa.3160410302.

[12]

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations, I. Existence of ground state,, Arch. Rational Mech. Anal., 82 (1983), 313.

[13]

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic problems involving critical Sovolev exponents,, Comm. Pure Appl. Math., 36 (1983), 437. doi: 10.1002/cpa.3160360405.

[14]

H. Brezis and S. Wainger, A note on limiting cases of Sobolev embeddings,, Comm. Partial Diff. Eqations, 5 (1980), 773. doi: 10.1080/03605308008820154.

[15]

D. M. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent in $\mathbb R^2$,, Comm. Partial Differential Equations, 17 (1992), 407. doi: 10.1080/03605309208820848.

[16]

L. Carleson and S.-Y. A. Chang, On the existence of an extremal function for an inequality of J. Moser,, Bull. Sci. Math., 110 (1986), 113.

[17]

M. Calanchi and E. Terraneo, Non-radial maximizers for functionals with exponential non-linearity in $\mathbb R^2$,, Adv. Nonlinear Stud., 5 (2005), 337.

[18]

P. Cherrier, Cas d'exception du théorème d'inclusion de Sobolev sur le variétés Riemanniennes e applications,, Bull. Sci. Math. (2), 105 (1981), 235.

[19]

A. Cianchi, A sharp embedding theorem for Orlicz-Sobolev spaces,, Indiana U. Math. J., 45 (1996), 39. doi: 10.1512/iumj.1996.45.1958.

[20]

A. Cianchi, Moser-Trudinger inequalities without boundary conditions and isoperimetric problems,, Indiana U. Math. J., 54 (2005), 669. doi: 10.1512/iumj.2005.54.2589.

[21]

A. Cianchi, Moser-Trudinger trace inequalities,, Adv. Math., 217 (2008), 2005. doi: 10.1016/j.aim.2007.09.007.

[22]

M. Comte, Solutions of elliptic equations with critical exponents in dimension 3,, Nonlin. Ana. TMA, 17 (1991), 445. doi: 10.1016/0362-546X(91)90139-R.

[23]

J.-M. Coron, Topologie et cas limite des injections de Sobolev, (French) [Topology and limit case of Sobolev embeddings],, C. R. Acad. Sc. Paris Ser. I, 299 (1984), 209.

[24]

D. G. de Figueiredo, Positive solutions of semilinear elliptic problems,, Differential equations (S\ Ao Paulo, 957 (1981), 34.

[25]

D. G. de Figueiredo and P. Felmer, On superquadratic elliptic systems,, Trans. Am. Math. Soc., 343 (1994), 99.

[26]

D. G. de Figueiredo, J. M. do Ó and B. Ruf, On an inequality by N. Trudinger and J. Moser and related elliptic equations,, Comm. Pure Appl. Math., 55 (2002), 135. doi: 10.1002/cpa.10015.

[27]

D. G. de Figueiredo, J. M. do Ó and B. Ruf, Critical and subcritical elliptic systems in dimension two,, Indiana Univ. Math. J., 53 (2004), 1037. doi: 10.1512/iumj.2004.53.2402.

[28]

D. G. de Figueiredo, O. H. Miyagaki and B. Ruf, Elliptic equations in $\mathbb R^2$ with nonlinearities in the critical growth range,, Calc. Var., 3 (1995), 139. doi: 10.1007/BF01205003.

[29]

D. G. de Figueiredo and B. Ruf, Existence and non-existence of radial solutions for elliptic equations with critical exponent in $\mathbb R^2$,, Comm. Pure Appl. Math., 48 (1995), 639. doi: 10.1002/cpa.3160480605.

[30]

M. del Pino, M. Musso and B. Ruf, New solutions for Trudinger-Moser critical equations in $\mathbb R^2$,, J. Functional Analysis, 258 (2010), 421. doi: 10.1016/j.jfa.2009.06.018.

[31]

J. M. do Ó, Semilinear Dirichlet problems for the $N-$Laplacian in $\mathbb R^N$ with nonlinearities in the critical growth range,, Differential Integral Equations, 9 (1996), 967.

[32]

J. M. do Ó, $N$-Laplacian equations in $ \mathbbR^N$ with critical growth,, Abstr. Appl. Anal., 2 (1997), 301. doi: 10.1155/S1085337597000419.

[33]

O. Druet, Multibump analysis in dimension 2: Quantification of blow-up levels,, Duke Math. J., 132 (2006), 217. doi: 10.1215/S0012-7094-06-13222-2.

[34]

D. E. Edmunds, P. Gurka and B. Opic, Double exponential integrability of convolution operators in generalized Lorentz-Zygmund spaces,, Indiana Univ. Math. J., 44 (1995), 19. doi: 10.1512/iumj.1995.44.1977.

[35]

M. Flucher, Extremal functions for the Trudinger-Moser inequality in 2 dimensions,, Comment. Math. Helv., 67 (1992), 471. doi: 10.1007/BF02566514.

[36]

L. Fontana, Sharp borderline Sobolev inequalities on compact Riemannian manifolds,, Comment. Math. Helvetici, 68 (1993), 415. doi: 10.1007/BF02565828.

[37]

N. Fusco, P.-L. Lions and C. Sbordone, Sobolev imbedding theorems in borderline cases,, Proc. AMS, 124 (1996), 561.

[38]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle,, Comm. Math. Phys., 68 (1979), 209. doi: 10.1007/BF01221125.

[39]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,'', Second Edition. Grundlehren der Mathematischen Wissenschaften, 224 (1983).

[40]

S. Hencl, A sharp form of an embedding into exponential and double exponential spaces,, J. Funct. A., 204 (2003), 196. doi: 10.1016/S0022-1236(02)00172-6.

[41]

J. Hulshof and R. van der Vorst, Differential systems with strongly indefinite variational structure,, J. Funct. Anal., 114 (1993), 32. doi: 10.1006/jfan.1993.1062.

[42]

J. Hulshof, E. Mitidieri and R. Van der Vorst, Strongly indefinite systems with critical Sobolev exponents., Trans. Amer. Math. Soc., 350 (1998), 2349. doi: 10.1090/S0002-9947-98-02159-X.

[43]

Y. Li, Moser-Trudinger inequality on compact Riemannian manifolds of dimension two,, J. Partial Differential Equ., 14 (2001), 163.

[44]

Y. Li, Extremal functions for the Moser-Trudinger inequalities on compact Riemannian manifolds,, Sci. China Ser. A, 48 (2005), 618. doi: 10.1360/04ys0050.

[45]

K. C. Lin, Extremal functions for Moser's inequality,, Trans. Amer. Math. Soc., 348 (1996), 2663. doi: 10.1090/S0002-9947-96-01541-3.

[46]

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case, part 1,, Rev. Mat. Iberoamericana, 1 (1985), 145.

[47]

G. G. Lorentz, On the theory of spaces $\Lambda$,, Pacific J. Math, 1 (1951), 411.

[48]

J. Moser, A sharp form of an inequality by N. Trudinger,, Indiana Univ. Math. J., 20 (): 1077. doi: 10.1512/iumj.1971.20.20101.

[49]

S. I. Pohozaev, The Sobolev embedding in the case $pl = n$,, Proc. of the Technical Scientific Conference on Advances of Scientific Research 1964-1965, (1965), 1964.

[50]

S. I. Pohozaev, Eigenfunctions of the equation $\Delta u + \lambda f(u) = 0$,, Dokl. Adad. Nauk SSSR, 165 (1965), 36.

[51]

P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations,'', CBMS Regional Conf. Ser. in Math., 65 (1986).

[52]

B. Ruf, Lorentz spaces and nonlinear elliptic systems,, Contributions to Nonlinear Analysis, 66 (2006), 471.

[53]

B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $\mathbb R^2$,, J. Funct. Analysis, 219 (2004), 340. doi: 10.1016/j.jfa.2004.06.013.

[54]

B. Ruf and C. Tarsi, On Trudinger-Moser type inequalities involving Sobolev-Lorentz spaces,, Annali Mat. Pura ed Appl. 1, 88 (2009), 369.

[55]

J. Sacks and K. Uhlenbeck, The existence of minimal immersions of 2-spheres,, Ann. Math., 113 (1981), 1. doi: 10.2307/1971131.

[56]

W. A. Strauss, Existence of solitary waves in higher dimensions,, Comm. Math. Phys., 55 (1977), 149. doi: 10.1007/BF01626517.

[57]

M. Struwe, Critical points of embeddings of $H^{1,n}_0$ into Orlicz spaces,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 5 (1988), 425.

[58]

M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities,, Math. Z., 187 (1984), 511. doi: 10.1007/BF01174186.

[59]

M. Struwe, Positive solutions of critical semilinear elliptic equations on non-contractible,, J. Eur. Math. Soc., 2 (2000), 329. doi: 10.1007/s100970000023.

[60]

N. S. Trudinger, On embeddings into Orlicz spaces and some applications,, J. Math. Mech., 17 (1967), 473.

[61]

Y. Yang, Extremal functions for Moser-Trudinger inequalities on 2-dimensional compact Riemannian manifolds with boundary,, Internat. J. Math., 17 (2006), 313. doi: 10.1142/S0129167X06003473.

show all references

References:
[1]

S. Adachi and K. Tanaka, Trudinger type inequalities in $\mathbbR^N$ and their best exponents,, Proc. Amer. Math. Soc., 128 (2000), 2051. doi: 10.1090/S0002-9939-99-05180-1.

[2]

D. R. Adams, A sharp inequality of J. Moser for higher order derivatives,, Annals of Math., 128 (1988), 385. doi: 10.2307/1971445.

[3]

R. A. Adams and J. F. Fournier, "Sobolev Spaces,'', Second Edition, (2003).

[4]

Adimurthi, Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the $n$-Laplacian,, Ann. Sc. Norm. Sup. Pisa, XVII (1990), 393.

[5]

Adimurthi and O. Druet, Blow-up analysis in dimension 2 and a sharp form of Trudinger-Moser inequality,, Comm. Part. Diff. Equ., 29 (2004), 295. doi: 10.1081/PDE-120028854.

[6]

Adimurthi and K. Sandeep, A singular Moser-Trudinger embedding and its applications,, NoDEA Nonlinear Differential Equations Appl., 13 (2007), 585. doi: 10.1007/s00030-006-4025-9.

[7]

A. Alvino, V. Ferone and G. Trombetti, Moser-type inequalities in Lorentz spaces,, Potential Anal., 5 (1996), 273.

[8]

A. Alvino, P.-L. Lions and G. Trombetti, On optimization problems with prescribed rearrangements,, Nonlinear Anal. T.M.A., 13 (1989), 185. doi: 10.1016/0362-546X(89)90043-6.

[9]

V. V. Atkinson and L. A. Peletier, Ground states and Dirichlet problems for $-\Delta u = f(u)$ in $\mathbb R^2$,, Arch. Rational Mech. Anal., 96 (1986), 147. doi: 10.1007/BF00251409.

[10]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications,, J. Functional Analysis, 14 (1973), 349. doi: 10.1016/0022-1236(73)90051-7.

[11]

A. Bahri and J.-M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: The effect of topology of the domain,, Comm. Pure Appl. Math., 41 (1988), 253. doi: 10.1002/cpa.3160410302.

[12]

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations, I. Existence of ground state,, Arch. Rational Mech. Anal., 82 (1983), 313.

[13]

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic problems involving critical Sovolev exponents,, Comm. Pure Appl. Math., 36 (1983), 437. doi: 10.1002/cpa.3160360405.

[14]

H. Brezis and S. Wainger, A note on limiting cases of Sobolev embeddings,, Comm. Partial Diff. Eqations, 5 (1980), 773. doi: 10.1080/03605308008820154.

[15]

D. M. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent in $\mathbb R^2$,, Comm. Partial Differential Equations, 17 (1992), 407. doi: 10.1080/03605309208820848.

[16]

L. Carleson and S.-Y. A. Chang, On the existence of an extremal function for an inequality of J. Moser,, Bull. Sci. Math., 110 (1986), 113.

[17]

M. Calanchi and E. Terraneo, Non-radial maximizers for functionals with exponential non-linearity in $\mathbb R^2$,, Adv. Nonlinear Stud., 5 (2005), 337.

[18]

P. Cherrier, Cas d'exception du théorème d'inclusion de Sobolev sur le variétés Riemanniennes e applications,, Bull. Sci. Math. (2), 105 (1981), 235.

[19]

A. Cianchi, A sharp embedding theorem for Orlicz-Sobolev spaces,, Indiana U. Math. J., 45 (1996), 39. doi: 10.1512/iumj.1996.45.1958.

[20]

A. Cianchi, Moser-Trudinger inequalities without boundary conditions and isoperimetric problems,, Indiana U. Math. J., 54 (2005), 669. doi: 10.1512/iumj.2005.54.2589.

[21]

A. Cianchi, Moser-Trudinger trace inequalities,, Adv. Math., 217 (2008), 2005. doi: 10.1016/j.aim.2007.09.007.

[22]

M. Comte, Solutions of elliptic equations with critical exponents in dimension 3,, Nonlin. Ana. TMA, 17 (1991), 445. doi: 10.1016/0362-546X(91)90139-R.

[23]

J.-M. Coron, Topologie et cas limite des injections de Sobolev, (French) [Topology and limit case of Sobolev embeddings],, C. R. Acad. Sc. Paris Ser. I, 299 (1984), 209.

[24]

D. G. de Figueiredo, Positive solutions of semilinear elliptic problems,, Differential equations (S\ Ao Paulo, 957 (1981), 34.

[25]

D. G. de Figueiredo and P. Felmer, On superquadratic elliptic systems,, Trans. Am. Math. Soc., 343 (1994), 99.

[26]

D. G. de Figueiredo, J. M. do Ó and B. Ruf, On an inequality by N. Trudinger and J. Moser and related elliptic equations,, Comm. Pure Appl. Math., 55 (2002), 135. doi: 10.1002/cpa.10015.

[27]

D. G. de Figueiredo, J. M. do Ó and B. Ruf, Critical and subcritical elliptic systems in dimension two,, Indiana Univ. Math. J., 53 (2004), 1037. doi: 10.1512/iumj.2004.53.2402.

[28]

D. G. de Figueiredo, O. H. Miyagaki and B. Ruf, Elliptic equations in $\mathbb R^2$ with nonlinearities in the critical growth range,, Calc. Var., 3 (1995), 139. doi: 10.1007/BF01205003.

[29]

D. G. de Figueiredo and B. Ruf, Existence and non-existence of radial solutions for elliptic equations with critical exponent in $\mathbb R^2$,, Comm. Pure Appl. Math., 48 (1995), 639. doi: 10.1002/cpa.3160480605.

[30]

M. del Pino, M. Musso and B. Ruf, New solutions for Trudinger-Moser critical equations in $\mathbb R^2$,, J. Functional Analysis, 258 (2010), 421. doi: 10.1016/j.jfa.2009.06.018.

[31]

J. M. do Ó, Semilinear Dirichlet problems for the $N-$Laplacian in $\mathbb R^N$ with nonlinearities in the critical growth range,, Differential Integral Equations, 9 (1996), 967.

[32]

J. M. do Ó, $N$-Laplacian equations in $ \mathbbR^N$ with critical growth,, Abstr. Appl. Anal., 2 (1997), 301. doi: 10.1155/S1085337597000419.

[33]

O. Druet, Multibump analysis in dimension 2: Quantification of blow-up levels,, Duke Math. J., 132 (2006), 217. doi: 10.1215/S0012-7094-06-13222-2.

[34]

D. E. Edmunds, P. Gurka and B. Opic, Double exponential integrability of convolution operators in generalized Lorentz-Zygmund spaces,, Indiana Univ. Math. J., 44 (1995), 19. doi: 10.1512/iumj.1995.44.1977.

[35]

M. Flucher, Extremal functions for the Trudinger-Moser inequality in 2 dimensions,, Comment. Math. Helv., 67 (1992), 471. doi: 10.1007/BF02566514.

[36]

L. Fontana, Sharp borderline Sobolev inequalities on compact Riemannian manifolds,, Comment. Math. Helvetici, 68 (1993), 415. doi: 10.1007/BF02565828.

[37]

N. Fusco, P.-L. Lions and C. Sbordone, Sobolev imbedding theorems in borderline cases,, Proc. AMS, 124 (1996), 561.

[38]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle,, Comm. Math. Phys., 68 (1979), 209. doi: 10.1007/BF01221125.

[39]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,'', Second Edition. Grundlehren der Mathematischen Wissenschaften, 224 (1983).

[40]

S. Hencl, A sharp form of an embedding into exponential and double exponential spaces,, J. Funct. A., 204 (2003), 196. doi: 10.1016/S0022-1236(02)00172-6.

[41]

J. Hulshof and R. van der Vorst, Differential systems with strongly indefinite variational structure,, J. Funct. Anal., 114 (1993), 32. doi: 10.1006/jfan.1993.1062.

[42]

J. Hulshof, E. Mitidieri and R. Van der Vorst, Strongly indefinite systems with critical Sobolev exponents., Trans. Amer. Math. Soc., 350 (1998), 2349. doi: 10.1090/S0002-9947-98-02159-X.

[43]

Y. Li, Moser-Trudinger inequality on compact Riemannian manifolds of dimension two,, J. Partial Differential Equ., 14 (2001), 163.

[44]

Y. Li, Extremal functions for the Moser-Trudinger inequalities on compact Riemannian manifolds,, Sci. China Ser. A, 48 (2005), 618. doi: 10.1360/04ys0050.

[45]

K. C. Lin, Extremal functions for Moser's inequality,, Trans. Amer. Math. Soc., 348 (1996), 2663. doi: 10.1090/S0002-9947-96-01541-3.

[46]

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case, part 1,, Rev. Mat. Iberoamericana, 1 (1985), 145.

[47]

G. G. Lorentz, On the theory of spaces $\Lambda$,, Pacific J. Math, 1 (1951), 411.

[48]

J. Moser, A sharp form of an inequality by N. Trudinger,, Indiana Univ. Math. J., 20 (): 1077. doi: 10.1512/iumj.1971.20.20101.

[49]

S. I. Pohozaev, The Sobolev embedding in the case $pl = n$,, Proc. of the Technical Scientific Conference on Advances of Scientific Research 1964-1965, (1965), 1964.

[50]

S. I. Pohozaev, Eigenfunctions of the equation $\Delta u + \lambda f(u) = 0$,, Dokl. Adad. Nauk SSSR, 165 (1965), 36.

[51]

P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations,'', CBMS Regional Conf. Ser. in Math., 65 (1986).

[52]

B. Ruf, Lorentz spaces and nonlinear elliptic systems,, Contributions to Nonlinear Analysis, 66 (2006), 471.

[53]

B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $\mathbb R^2$,, J. Funct. Analysis, 219 (2004), 340. doi: 10.1016/j.jfa.2004.06.013.

[54]

B. Ruf and C. Tarsi, On Trudinger-Moser type inequalities involving Sobolev-Lorentz spaces,, Annali Mat. Pura ed Appl. 1, 88 (2009), 369.

[55]

J. Sacks and K. Uhlenbeck, The existence of minimal immersions of 2-spheres,, Ann. Math., 113 (1981), 1. doi: 10.2307/1971131.

[56]

W. A. Strauss, Existence of solitary waves in higher dimensions,, Comm. Math. Phys., 55 (1977), 149. doi: 10.1007/BF01626517.

[57]

M. Struwe, Critical points of embeddings of $H^{1,n}_0$ into Orlicz spaces,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 5 (1988), 425.

[58]

M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities,, Math. Z., 187 (1984), 511. doi: 10.1007/BF01174186.

[59]

M. Struwe, Positive solutions of critical semilinear elliptic equations on non-contractible,, J. Eur. Math. Soc., 2 (2000), 329. doi: 10.1007/s100970000023.

[60]

N. S. Trudinger, On embeddings into Orlicz spaces and some applications,, J. Math. Mech., 17 (1967), 473.

[61]

Y. Yang, Extremal functions for Moser-Trudinger inequalities on 2-dimensional compact Riemannian manifolds with boundary,, Internat. J. Math., 17 (2006), 313. doi: 10.1142/S0129167X06003473.

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