Elliptic equations and systems with critical Trudinger-Moser nonlinearities

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

Elliptic equations and systems with critical Trudinger-Moser nonlinearities

Djairo G. De Figueiredo ^1, , João Marcos do Ó ^2, and Bernhard Ruf ^3,

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

Abstract
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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.

Keywords: variational methods, Trudinger-Moser inequality, critical growth..

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.

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. Google Scholar
[2]	D. R. Adams, A sharp inequality of J. Moser for higher order derivatives,, Annals of Math., 128 (1988), 385. doi: 10.2307/1971445. Google Scholar
[3]	R. A. Adams and J. F. Fournier, "Sobolev Spaces,'', Second Edition, (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, XVII (1990), 393. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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., 110 (1986), 113. 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. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[21]	A. Cianchi, Moser-Trudinger trace inequalities,, Adv. Math., 217 (2008), 2005. 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. 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. Google Scholar
[24]	D. G. de Figueiredo, Positive solutions of semilinear elliptic problems,, Differential equations (S\ Ao Paulo, 957 (1981), 34. Google Scholar
[25]	D. G. de Figueiredo and P. Felmer, On superquadratic elliptic systems,, Trans. Am. Math. Soc., 343 (1994), 99. 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. 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. 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. 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. 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. 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. Google Scholar
[32]	J. M. do Ó, $N$-Laplacian equations in $ \mathbbR^N$ with critical growth,, Abstr. Appl. Anal., 2 (1997), 301. 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. 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. 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. doi: 10.1007/BF02566514. Google Scholar
[36]	L. Fontana, Sharp borderline Sobolev inequalities on compact Riemannian manifolds,, Comment. Math. Helvetici, 68 (1993), 415. 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. 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. 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 (1983). Google Scholar
[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. Google Scholar
[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. 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. 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. Google Scholar
[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. Google Scholar
[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. 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. Google Scholar
[47]	G. G. Lorentz, On the theory of spaces $\Lambda$,, Pacific J. Math, 1 (1951), 411. 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, (1965), 1964. Google Scholar
[50]	S. I. Pohozaev, Eigenfunctions of the equation $\Delta u + \lambda f(u) = 0$,, Dokl. Adad. Nauk SSSR, 165 (1965), 36. Google Scholar
[51]	P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations,'', CBMS Regional Conf. Ser. in Math., 65 (1986). Google Scholar
[52]	B. Ruf, Lorentz spaces and nonlinear elliptic systems,, Contributions to Nonlinear Analysis, 66 (2006), 471. Google Scholar
[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. 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. Google Scholar
[55]	J. Sacks and K. Uhlenbeck, The existence of minimal immersions of 2-spheres,, Ann. Math., 113 (1981), 1. doi: 10.2307/1971131. Google Scholar
[56]	W. A. Strauss, Existence of solitary waves in higher dimensions,, Comm. Math. Phys., 55 (1977), 149. doi: 10.1007/BF01626517. Google Scholar
[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. Google Scholar
[58]	M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities,, Math. Z., 187 (1984), 511. 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. doi: 10.1007/s100970000023. Google Scholar
[60]	N. S. Trudinger, On embeddings into Orlicz spaces and some applications,, J. Math. Mech., 17 (1967), 473. 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. 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. 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. doi: 10.2307/1971445. Google Scholar
[3]	R. A. Adams and J. F. Fournier, "Sobolev Spaces,'', Second Edition, (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, XVII (1990), 393. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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., 110 (1986), 113. 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. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[21]	A. Cianchi, Moser-Trudinger trace inequalities,, Adv. Math., 217 (2008), 2005. 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. 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. Google Scholar
[24]	D. G. de Figueiredo, Positive solutions of semilinear elliptic problems,, Differential equations (S\ Ao Paulo, 957 (1981), 34. Google Scholar
[25]	D. G. de Figueiredo and P. Felmer, On superquadratic elliptic systems,, Trans. Am. Math. Soc., 343 (1994), 99. 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. 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. 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. 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. 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. 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. Google Scholar
[32]	J. M. do Ó, $N$-Laplacian equations in $ \mathbbR^N$ with critical growth,, Abstr. Appl. Anal., 2 (1997), 301. 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. 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. 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. doi: 10.1007/BF02566514. Google Scholar
[36]	L. Fontana, Sharp borderline Sobolev inequalities on compact Riemannian manifolds,, Comment. Math. Helvetici, 68 (1993), 415. 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. 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. 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 (1983). Google Scholar
[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. Google Scholar
[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. 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. 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. Google Scholar
[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. Google Scholar
[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. 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. Google Scholar
[47]	G. G. Lorentz, On the theory of spaces $\Lambda$,, Pacific J. Math, 1 (1951), 411. 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, (1965), 1964. Google Scholar
[50]	S. I. Pohozaev, Eigenfunctions of the equation $\Delta u + \lambda f(u) = 0$,, Dokl. Adad. Nauk SSSR, 165 (1965), 36. Google Scholar
[51]	P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations,'', CBMS Regional Conf. Ser. in Math., 65 (1986). Google Scholar
[52]	B. Ruf, Lorentz spaces and nonlinear elliptic systems,, Contributions to Nonlinear Analysis, 66 (2006), 471. Google Scholar
[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. 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. Google Scholar
[55]	J. Sacks and K. Uhlenbeck, The existence of minimal immersions of 2-spheres,, Ann. Math., 113 (1981), 1. doi: 10.2307/1971131. Google Scholar
[56]	W. A. Strauss, Existence of solitary waves in higher dimensions,, Comm. Math. Phys., 55 (1977), 149. doi: 10.1007/BF01626517. Google Scholar
[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. Google Scholar
[58]	M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities,, Math. Z., 187 (1984), 511. 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. doi: 10.1007/s100970000023. Google Scholar
[60]	N. S. Trudinger, On embeddings into Orlicz spaces and some applications,, J. Math. Mech., 17 (1967), 473. 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. doi: 10.1142/S0129167X06003473. Google Scholar

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