May  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, 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), 2051-2057. doi: 10.1090/S0002-9939-99-05180-1.  Google Scholar

[2]

D. R. Adams, A sharp inequality of J. Moser for higher order derivatives, Annals of Math., 128 (1988), 385-398. doi: 10.2307/1971445.  Google Scholar

[3]

R. A. Adams and J. F. Fournier, "Sobolev Spaces,'' Second Edition, Academic Press, 2003  Google Scholar

[4]

Adimurthi, Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the $n$-Laplacian, Ann. Sc. Norm. Sup. Pisa, vol XVII (1990), 393-413.  Google Scholar

[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), 295-322. doi: 10.1081/PDE-120028854.  Google Scholar

[6]

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.  Google Scholar

[7]

A. Alvino, V. Ferone and G. Trombetti, Moser-type inequalities in Lorentz spaces, Potential Anal., 5 (1996), 273-299.  Google Scholar

[8]

A. Alvino, P.-L. Lions and G. Trombetti, On optimization problems with prescribed rearrangements, Nonlinear Anal. T.M.A., 13 (1989), 185-220. doi: 10.1016/0362-546X(89)90043-6.  Google Scholar

[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), 147-165. doi: 10.1007/BF00251409.  Google Scholar

[10]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381. doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[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), 253-294. doi: 10.1002/cpa.3160410302.  Google Scholar

[12]

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations, I. Existence of ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345.  Google Scholar

[13]

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic problems involving critical Sovolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405.  Google Scholar

[14]

H. Brezis and S. Wainger, A note on limiting cases of Sobolev embeddings, Comm. Partial Diff. Eqations, 5 (1980), 773-789. doi: 10.1080/03605308008820154.  Google Scholar

[15]

D. M. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent in $\mathbb R^2$, Comm. Partial Differential Equations, 17 (1992), 407-435. doi: 10.1080/03605309208820848.  Google Scholar

[16]

L. Carleson and S.-Y. A. Chang, On the existence of an extremal function for an inequality of J. Moser, Bull. Sci. Math., Sér. 2, 110 (1986), 113-127.  Google Scholar

[17]

M. Calanchi and E. Terraneo, Non-radial maximizers for functionals with exponential non-linearity in $\mathbb R^2$, Adv. Nonlinear Stud., 5 (2005), 337-350.  Google Scholar

[18]

P. Cherrier, Cas d'exception du théorème d'inclusion de Sobolev sur le variétés Riemanniennes e applications, Bull. Sci. Math. (2), 105 (1981), 235-288.  Google Scholar

[19]

A. Cianchi, A sharp embedding theorem for Orlicz-Sobolev spaces, Indiana U. Math. J., 45 (1996), 39-65. doi: 10.1512/iumj.1996.45.1958.  Google Scholar

[20]

A. Cianchi, Moser-Trudinger inequalities without boundary conditions and isoperimetric problems, Indiana U. Math. J., 54 (2005), 669-705. doi: 10.1512/iumj.2005.54.2589.  Google Scholar

[21]

A. Cianchi, Moser-Trudinger trace inequalities, Adv. Math., 217 (2008), 2005-2044. doi: 10.1016/j.aim.2007.09.007.  Google Scholar

[22]

M. Comte, Solutions of elliptic equations with critical exponents in dimension 3, Nonlin. Ana. TMA, 17 (1991), 445-455 doi: 10.1016/0362-546X(91)90139-R.  Google Scholar

[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), 209-212.  Google Scholar

[24]

D. G. de Figueiredo, Positive solutions of semilinear elliptic problems, Differential equations (SÃo Paulo, 1981), pp. 34-87, Lecture Notes in Math., 957, Springer, Berlin-New York, 1982. Google Scholar

[25]

D. G. de Figueiredo and P. Felmer, On superquadratic elliptic systems, Trans. Am. Math. Soc., 343 (1994), 99-116.  Google Scholar

[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), 135-152. doi: 10.1002/cpa.10015.  Google Scholar

[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), 1037-1054. doi: 10.1512/iumj.2004.53.2402.  Google Scholar

[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), 139-153. doi: 10.1007/BF01205003.  Google Scholar

[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), 639-655 doi: 10.1002/cpa.3160480605.  Google Scholar

[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), 421-457. doi: 10.1016/j.jfa.2009.06.018.  Google Scholar

[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), 967-979.  Google Scholar

[32]

J. M. do Ó, $N$-Laplacian equations in $ \mathbbR^N$ with critical growth, Abstr. Appl. Anal., 2 (1997), 301-315. doi: 10.1155/S1085337597000419.  Google Scholar

[33]

O. Druet, Multibump analysis in dimension 2: Quantification of blow-up levels, Duke Math. J., 132 (2006), 217-269. doi: 10.1215/S0012-7094-06-13222-2.  Google Scholar

[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), 19-43. doi: 10.1512/iumj.1995.44.1977.  Google Scholar

[35]

M. Flucher, Extremal functions for the Trudinger-Moser inequality in 2 dimensions, Comment. Math. Helv., 67 (1992), 471-497. doi: 10.1007/BF02566514.  Google Scholar

[36]

L. Fontana, Sharp borderline Sobolev inequalities on compact Riemannian manifolds, Comment. Math. Helvetici, 68 (1993), 415-454. doi: 10.1007/BF02565828.  Google Scholar

[37]

N. Fusco, P.-L. Lions and C. Sbordone, Sobolev imbedding theorems in borderline cases, Proc. AMS, 124 (1996), 561-565.  Google Scholar

[38]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. doi: 10.1007/BF01221125.  Google Scholar

[39]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,'' Second Edition. Grundlehren der Mathematischen Wissenschaften, 224 Springer-Verlag, Berlin, 1983.  Google Scholar

[40]

S. Hencl, A sharp form of an embedding into exponential and double exponential spaces, J. Funct. A., 204 (2003), 196-227. doi: 10.1016/S0022-1236(02)00172-6.  Google Scholar

[41]

J. Hulshof and R. van der Vorst, Differential systems with strongly indefinite variational structure, J. Funct. Anal., 114 (1993), 32-58. doi: 10.1006/jfan.1993.1062.  Google Scholar

[42]

J. Hulshof, E. Mitidieri and R. Van der Vorst, Strongly indefinite systems with critical Sobolev exponents. Trans. Amer. Math. Soc., 350 (1998), 2349-2365. doi: 10.1090/S0002-9947-98-02159-X.  Google Scholar

[43]

Y. Li, Moser-Trudinger inequality on compact Riemannian manifolds of dimension two, J. Partial Differential Equ., 14 (2001), 163-192.  Google Scholar

[44]

Y. Li, Extremal functions for the Moser-Trudinger inequalities on compact Riemannian manifolds, Sci. China Ser. A, 48 (2005), 618-648. doi: 10.1360/04ys0050.  Google Scholar

[45]

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.  Google Scholar

[46]

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case, part 1, Rev. Mat. Iberoamericana, 1 (1985), 145-201.  Google Scholar

[47]

G. G. Lorentz, On the theory of spaces $\Lambda$, Pacific J. Math, 1 (1951), 411-429.  Google Scholar

[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.  Google Scholar

[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, Mathematics Section, (Moskov. Energet. Inst., Moscow), (1965), 158-170. Google Scholar

[50]

S. I. Pohozaev, Eigenfunctions of the equation $\Delta u + \lambda f(u) = 0$, Dokl. Adad. Nauk SSSR, 165 (1965), 36-39. [Sov. Math., Dokl. 6, 1408-1411 (1965)] Google Scholar

[51]

P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations,'' CBMS Regional Conf. Ser. in Math., 65, AMS, Providence, RI, 1986.  Google Scholar

[52]

B. Ruf, Lorentz spaces and nonlinear elliptic systems, Contributions to Nonlinear Analysis, Progr. Nonlinear Differential Equations Appl., 66, Birkhäuser, Basel, (2006), 471-489.  Google Scholar

[53]

B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $\mathbb R^2$, J. Funct. Analysis, 219 (2004), 340-367. doi: 10.1016/j.jfa.2004.06.013.  Google Scholar

[54]

B. Ruf and C. Tarsi, On Trudinger-Moser type inequalities involving Sobolev-Lorentz spaces, Annali Mat. Pura ed Appl. 1, 88 (2009), 369-397.  Google Scholar

[55]

J. Sacks and K. Uhlenbeck, The existence of minimal immersions of 2-spheres, Ann. Math., 113 (1981), 1-24. doi: 10.2307/1971131.  Google Scholar

[56]

W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162. doi: 10.1007/BF01626517.  Google Scholar

[57]

M. Struwe, Critical points of embeddings of $H^{1,n}_0$ into Orlicz spaces, Ann. Inst. H. Poincaré Anal. Non Linéaire, 5 (1988), 425-464.  Google Scholar

[58]

M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z., 187 (1984), 511-517. doi: 10.1007/BF01174186.  Google Scholar

[59]

M. Struwe, Positive solutions of critical semilinear elliptic equations on non-contractible, J. Eur. Math. Soc., 2 (2000), 329-388. doi: 10.1007/s100970000023.  Google Scholar

[60]

N. S. Trudinger, On embeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-483.  Google Scholar

[61]

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.  Google Scholar

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), 2051-2057. doi: 10.1090/S0002-9939-99-05180-1.  Google Scholar

[2]

D. R. Adams, A sharp inequality of J. Moser for higher order derivatives, Annals of Math., 128 (1988), 385-398. doi: 10.2307/1971445.  Google Scholar

[3]

R. A. Adams and J. F. Fournier, "Sobolev Spaces,'' Second Edition, Academic Press, 2003  Google Scholar

[4]

Adimurthi, Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the $n$-Laplacian, Ann. Sc. Norm. Sup. Pisa, vol XVII (1990), 393-413.  Google Scholar

[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), 295-322. doi: 10.1081/PDE-120028854.  Google Scholar

[6]

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.  Google Scholar

[7]

A. Alvino, V. Ferone and G. Trombetti, Moser-type inequalities in Lorentz spaces, Potential Anal., 5 (1996), 273-299.  Google Scholar

[8]

A. Alvino, P.-L. Lions and G. Trombetti, On optimization problems with prescribed rearrangements, Nonlinear Anal. T.M.A., 13 (1989), 185-220. doi: 10.1016/0362-546X(89)90043-6.  Google Scholar

[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), 147-165. doi: 10.1007/BF00251409.  Google Scholar

[10]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381. doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[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), 253-294. doi: 10.1002/cpa.3160410302.  Google Scholar

[12]

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations, I. Existence of ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345.  Google Scholar

[13]

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic problems involving critical Sovolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405.  Google Scholar

[14]

H. Brezis and S. Wainger, A note on limiting cases of Sobolev embeddings, Comm. Partial Diff. Eqations, 5 (1980), 773-789. doi: 10.1080/03605308008820154.  Google Scholar

[15]

D. M. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent in $\mathbb R^2$, Comm. Partial Differential Equations, 17 (1992), 407-435. doi: 10.1080/03605309208820848.  Google Scholar

[16]

L. Carleson and S.-Y. A. Chang, On the existence of an extremal function for an inequality of J. Moser, Bull. Sci. Math., Sér. 2, 110 (1986), 113-127.  Google Scholar

[17]

M. Calanchi and E. Terraneo, Non-radial maximizers for functionals with exponential non-linearity in $\mathbb R^2$, Adv. Nonlinear Stud., 5 (2005), 337-350.  Google Scholar

[18]

P. Cherrier, Cas d'exception du théorème d'inclusion de Sobolev sur le variétés Riemanniennes e applications, Bull. Sci. Math. (2), 105 (1981), 235-288.  Google Scholar

[19]

A. Cianchi, A sharp embedding theorem for Orlicz-Sobolev spaces, Indiana U. Math. J., 45 (1996), 39-65. doi: 10.1512/iumj.1996.45.1958.  Google Scholar

[20]

A. Cianchi, Moser-Trudinger inequalities without boundary conditions and isoperimetric problems, Indiana U. Math. J., 54 (2005), 669-705. doi: 10.1512/iumj.2005.54.2589.  Google Scholar

[21]

A. Cianchi, Moser-Trudinger trace inequalities, Adv. Math., 217 (2008), 2005-2044. doi: 10.1016/j.aim.2007.09.007.  Google Scholar

[22]

M. Comte, Solutions of elliptic equations with critical exponents in dimension 3, Nonlin. Ana. TMA, 17 (1991), 445-455 doi: 10.1016/0362-546X(91)90139-R.  Google Scholar

[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), 209-212.  Google Scholar

[24]

D. G. de Figueiredo, Positive solutions of semilinear elliptic problems, Differential equations (SÃo Paulo, 1981), pp. 34-87, Lecture Notes in Math., 957, Springer, Berlin-New York, 1982. Google Scholar

[25]

D. G. de Figueiredo and P. Felmer, On superquadratic elliptic systems, Trans. Am. Math. Soc., 343 (1994), 99-116.  Google Scholar

[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), 135-152. doi: 10.1002/cpa.10015.  Google Scholar

[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), 1037-1054. doi: 10.1512/iumj.2004.53.2402.  Google Scholar

[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), 139-153. doi: 10.1007/BF01205003.  Google Scholar

[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), 639-655 doi: 10.1002/cpa.3160480605.  Google Scholar

[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), 421-457. doi: 10.1016/j.jfa.2009.06.018.  Google Scholar

[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), 967-979.  Google Scholar

[32]

J. M. do Ó, $N$-Laplacian equations in $ \mathbbR^N$ with critical growth, Abstr. Appl. Anal., 2 (1997), 301-315. doi: 10.1155/S1085337597000419.  Google Scholar

[33]

O. Druet, Multibump analysis in dimension 2: Quantification of blow-up levels, Duke Math. J., 132 (2006), 217-269. doi: 10.1215/S0012-7094-06-13222-2.  Google Scholar

[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), 19-43. doi: 10.1512/iumj.1995.44.1977.  Google Scholar

[35]

M. Flucher, Extremal functions for the Trudinger-Moser inequality in 2 dimensions, Comment. Math. Helv., 67 (1992), 471-497. doi: 10.1007/BF02566514.  Google Scholar

[36]

L. Fontana, Sharp borderline Sobolev inequalities on compact Riemannian manifolds, Comment. Math. Helvetici, 68 (1993), 415-454. doi: 10.1007/BF02565828.  Google Scholar

[37]

N. Fusco, P.-L. Lions and C. Sbordone, Sobolev imbedding theorems in borderline cases, Proc. AMS, 124 (1996), 561-565.  Google Scholar

[38]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. doi: 10.1007/BF01221125.  Google Scholar

[39]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,'' Second Edition. Grundlehren der Mathematischen Wissenschaften, 224 Springer-Verlag, Berlin, 1983.  Google Scholar

[40]

S. Hencl, A sharp form of an embedding into exponential and double exponential spaces, J. Funct. A., 204 (2003), 196-227. doi: 10.1016/S0022-1236(02)00172-6.  Google Scholar

[41]

J. Hulshof and R. van der Vorst, Differential systems with strongly indefinite variational structure, J. Funct. Anal., 114 (1993), 32-58. doi: 10.1006/jfan.1993.1062.  Google Scholar

[42]

J. Hulshof, E. Mitidieri and R. Van der Vorst, Strongly indefinite systems with critical Sobolev exponents. Trans. Amer. Math. Soc., 350 (1998), 2349-2365. doi: 10.1090/S0002-9947-98-02159-X.  Google Scholar

[43]

Y. Li, Moser-Trudinger inequality on compact Riemannian manifolds of dimension two, J. Partial Differential Equ., 14 (2001), 163-192.  Google Scholar

[44]

Y. Li, Extremal functions for the Moser-Trudinger inequalities on compact Riemannian manifolds, Sci. China Ser. A, 48 (2005), 618-648. doi: 10.1360/04ys0050.  Google Scholar

[45]

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.  Google Scholar

[46]

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case, part 1, Rev. Mat. Iberoamericana, 1 (1985), 145-201.  Google Scholar

[47]

G. G. Lorentz, On the theory of spaces $\Lambda$, Pacific J. Math, 1 (1951), 411-429.  Google Scholar

[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.  Google Scholar

[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, Mathematics Section, (Moskov. Energet. Inst., Moscow), (1965), 158-170. Google Scholar

[50]

S. I. Pohozaev, Eigenfunctions of the equation $\Delta u + \lambda f(u) = 0$, Dokl. Adad. Nauk SSSR, 165 (1965), 36-39. [Sov. Math., Dokl. 6, 1408-1411 (1965)] Google Scholar

[51]

P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations,'' CBMS Regional Conf. Ser. in Math., 65, AMS, Providence, RI, 1986.  Google Scholar

[52]

B. Ruf, Lorentz spaces and nonlinear elliptic systems, Contributions to Nonlinear Analysis, Progr. Nonlinear Differential Equations Appl., 66, Birkhäuser, Basel, (2006), 471-489.  Google Scholar

[53]

B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $\mathbb R^2$, J. Funct. Analysis, 219 (2004), 340-367. doi: 10.1016/j.jfa.2004.06.013.  Google Scholar

[54]

B. Ruf and C. Tarsi, On Trudinger-Moser type inequalities involving Sobolev-Lorentz spaces, Annali Mat. Pura ed Appl. 1, 88 (2009), 369-397.  Google Scholar

[55]

J. Sacks and K. Uhlenbeck, The existence of minimal immersions of 2-spheres, Ann. Math., 113 (1981), 1-24. doi: 10.2307/1971131.  Google Scholar

[56]

W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162. doi: 10.1007/BF01626517.  Google Scholar

[57]

M. Struwe, Critical points of embeddings of $H^{1,n}_0$ into Orlicz spaces, Ann. Inst. H. Poincaré Anal. Non Linéaire, 5 (1988), 425-464.  Google Scholar

[58]

M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z., 187 (1984), 511-517. doi: 10.1007/BF01174186.  Google Scholar

[59]

M. Struwe, Positive solutions of critical semilinear elliptic equations on non-contractible, J. Eur. Math. Soc., 2 (2000), 329-388. doi: 10.1007/s100970000023.  Google Scholar

[60]

N. S. Trudinger, On embeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-483.  Google Scholar

[61]

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.  Google Scholar

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