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October  2020, 19(10): 4771-4796. doi: 10.3934/cpaa.2020211

On some elliptic equation in the whole euclidean space $ \mathbb{R}^2 $ with nonlinearities having new exponential growth condition

Sami Aouaoui 1,, and  Rahma Jlel 2,

1. 

Institut Supérieur des Mathématiques Appliquées et de, l'Informatique de Kairouan, Avenue Assad Iben Fourat, Kairouan, 3100, Tunisie
2. 

Faculté des Sciences de Monastir, Avenue de l'environnement 5019 Monastir, Tunisie

* Corresponding author

Received  May 2019 Revised  May 2020 Published  July 2020

In this paper, we study some elliptic equation defined in $ \mathbb{R}^2 $ and involving a nonlinearity with new exponential growth condition including the doubly exponential growth at infinity. For that aim, we start by extending some new Trudinger-Moser type inequalities defined on the unit ball of different classes of weighted Sobolev spaces established by B. Ruf and M. Calanchi to the whole space $ \mathbb{R}^2. $

Keywords: Trudinger-Moser inequality, doubly exponential, radial function, ground state.
Mathematics Subject Classification: Primary: 35A15, 35D30, 35J20; Secondary: 35J61.
Citation: Sami Aouaoui, Rahma Jlel. On some elliptic equation in the whole euclidean space $ \mathbb{R}^2 $ with nonlinearities having new exponential growth condition. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4771-4796. doi: 10.3934/cpaa.2020211
References:
[1]

Adimurthi and K. Sandeep, A singular Moser-Trudinger embedding and its applications, NoDea Nonlinear Differ. Equ. Appl., 13 (2007), 585-603.  doi: 10.1007/s00030-006-4025-9.  Google Scholar

[2]

Adimurthi and Y. Yang, An interpolation of Hardy inequality and Trudinger-Moser inequality in $\mathbb{R}^N$ and its applications, Int. Math. Res. Not. IMRN, 13 (2010), 2394-2426.  doi: 10.1093/imrn/rnp194.  Google Scholar

[3]

F. S. B. Albuquerque, C. O. Alves and E. S. Medeiros, Nonlinear Schrödinger equation with unbounded or decaying radial potentials involving exponential critical growth in $\mathbb{R}^2$, J. Math. Anal. Appl., 409 (2014), 1021-1031.  doi: 10.1016/j.jmaa.2013.07.005.  Google Scholar

[4]

F. S. B. Albuquerque, Sharp constant and extremal function for weighted Trudinger-Moser type inequalities in $\mathbb{R}^2$, J. Math. Anal. Appl., 421 (2015), 963-970.  doi: 10.1016/j.jmaa.2014.07.035.  Google Scholar

[5]

C. O. Alves, M. A. S. Souto and M. Montenegro, Existence of a ground state solution for a nonlinear scalar field equation with critical growth, Calc. Var. Partial Differ. Equ., 4 (2012), 537-554.  doi: 10.1007/s00526-011-0422-y.  Google Scholar

[6]

C. O. Alves, D. Cassani, C. Tarsi and M. Yang, Existence and concentration of ground state solutions for a critical nonlocal Schrödinger equation in $ \mathbb{R}^2 $, J. Differ. Equ., 261 (2016), 1933-1972.  doi: 10.1016/j.jde.2016.04.021.  Google Scholar

[7]

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

[8]

S. Aouaoui and F. S. B. Albuquerque, A weighted Trudinger-Moser type inequality and its applications to quasilinear elliptic problems with critical growth in the whole Euclidean space, Topol. Meth. Nonlinear Anal., 54 (2019), 109-130.  doi: 10.12775/tmna.2019.027.  Google Scholar

[9]

M. Calanchi, Some weighted inequalities of Trudinger-Moser Type. In: Analysis and Topology in Nonlinear Differential Equations, in Progress in Nonlinear Differential Equations and Applications, Springer, Birkhauser, Vol. 85, (2014), 163–174.  Google Scholar

[10]

M. Calanchi and B. Ruf, On Trudinger-Moser type inequalities with logarithmic weights, J. Differ. Equ., 258 (2015), 1967-1989.  doi: 10.1016/j.jde.2014.11.019.  Google Scholar

[11]

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.  Google Scholar

[12]

M. Calanchi, B. Ruf and F. Sani, Elliptic equations in dimension 2 with double exponential nonlinearities, NoDea Nonlinear Differ. Equ. Appl., 24 (2017), Art. 29. doi: 10.1007/s00030-017-0453-y.  Google Scholar

[13]

M. Calanchi and E. Terraneo, Non-radial maximizers for functionals with exponential nonlinearity in $\mathbb{R}^2$, Adv. Nonlinear Stud., 5 (2005), 337-350.  doi: 10.1515/ans-2005-0302.  Google Scholar

[14]

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

[15]

A. C. Cavalheiro, Weighted sobolev spaces and degenerate elliptic equations, Bol. Soc. Paran. Mat., 26 (2008), 117-132.  doi: 10.5269/bspm.v26i1-2.7415.  Google Scholar

[16]

J. F. De Oliveira and J.M. do Ò, Trudinger-Moser type inequalities for weighted spaces involving fractional dimensions, Proc. Amer. Math. Soc., 142 (2014), 2813-2828.  doi: 10.1090/S0002-9939-2014-12019-3.  Google Scholar

[17]

J. M. do Ò, N-Laplacian equations in $\mathbb{R}^N$ with critical growth, Abstr. Appl. Anal., 222 (1997), 301-315.  doi: 10.1155/S1085337597000419.  Google Scholar

[18]

J. M. do Ò and M. de Souza, On a class of singular Trudinger-Moser inequalities, Math. Nachr., 284 (2011), 1754-1776.  doi: 10.1002/mana.201000083.  Google Scholar

[19]

D. E. Edmunds, H. Hudzik and M. Krbec, On weighted critical imbeddings of Sobolev spaces, Math. Z., 286 (2011), 585-592.  doi: 10.1007/s00209-010-0684-7.  Google Scholar

[20]

M. F. Furtado, E. S. Medeiros and U. B. Severo, A Trudinger-Moser inequality in a weighted Sobolev space and applications, Math. Nach., 287 (2014), 1255-1273.  doi: 10.1002/mana.201200315.  Google Scholar

[21]

S. Goyal and K. Sreenadh, The Nehari manifold approach for $ N-$Laplace equation with singular and exponential nonlinearities in $ \mathbb{R}^2 $, Commun. Contemp. Math., (2014), Art. 1450011. doi: 10.1142/S0219199714500114.  Google Scholar

[22]

T. Kilpeläinen, Weighted Sobolev spaces and capacity, Ann. Acad. Sci. Fenn. Math., 19 (1994), 95-113.   Google Scholar

[23]

N. Lam, Sharp Trudinger-Moser inequalities with monomial weights, Nonlinear Differ. Equ. Appl., 24 (2017), Art. 39. doi: 10.1007/s00030-017-0456-8.  Google Scholar

[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.  Google Scholar

[25]

E. Nakai, N. Tomita and K. Yabuta, Density of the set of all infinitely differentiable functions with compact support in weighted Sobolev spaces, Sc. Math. Jpn., 10 (2004), 39-45.   Google Scholar

[26]

P. Pucci and V. Radulescu, The Impact of the Mountain Pass Theory in Nonlinear Analysis: a Mathematical Survey, Bollettino dell'Unione Matematica Italiana Serie, 9 (2010), 543-582.   Google Scholar

[27]

B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $\mathbb{R}^2$, J. Funct. Anal., 219 (2005), 340-367.  doi: 10.1016/j.jfa.2004.06.013.  Google Scholar

[28]

B. Ruf and F. Sani, Ground states for elliptic equations in $ \mathbb{R}^2 $ with exponential critical growth, in Geometric Properties for Parabolic and Elliptic PDE'S, Springer Serie, Vol. 2,251–268. doi: 10.1007/978-88-470-2841-8_16.  Google Scholar

[29]

W. A. Strauss, Existence of solitary waves in higher dimensions, Commun. Math. Phys., 55 (1977), 149-162.   Google Scholar

[30]

N. S. Trudinger, On the embedding into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-484.  doi: 10.1512/iumj.1968.17.17028.  Google Scholar

show all references

References:
[1]

Adimurthi and K. Sandeep, A singular Moser-Trudinger embedding and its applications, NoDea Nonlinear Differ. Equ. Appl., 13 (2007), 585-603.  doi: 10.1007/s00030-006-4025-9.  Google Scholar

[2]

Adimurthi and Y. Yang, An interpolation of Hardy inequality and Trudinger-Moser inequality in $\mathbb{R}^N$ and its applications, Int. Math. Res. Not. IMRN, 13 (2010), 2394-2426.  doi: 10.1093/imrn/rnp194.  Google Scholar

[3]

F. S. B. Albuquerque, C. O. Alves and E. S. Medeiros, Nonlinear Schrödinger equation with unbounded or decaying radial potentials involving exponential critical growth in $\mathbb{R}^2$, J. Math. Anal. Appl., 409 (2014), 1021-1031.  doi: 10.1016/j.jmaa.2013.07.005.  Google Scholar

[4]

F. S. B. Albuquerque, Sharp constant and extremal function for weighted Trudinger-Moser type inequalities in $\mathbb{R}^2$, J. Math. Anal. Appl., 421 (2015), 963-970.  doi: 10.1016/j.jmaa.2014.07.035.  Google Scholar

[5]

C. O. Alves, M. A. S. Souto and M. Montenegro, Existence of a ground state solution for a nonlinear scalar field equation with critical growth, Calc. Var. Partial Differ. Equ., 4 (2012), 537-554.  doi: 10.1007/s00526-011-0422-y.  Google Scholar

[6]

C. O. Alves, D. Cassani, C. Tarsi and M. Yang, Existence and concentration of ground state solutions for a critical nonlocal Schrödinger equation in $ \mathbb{R}^2 $, J. Differ. Equ., 261 (2016), 1933-1972.  doi: 10.1016/j.jde.2016.04.021.  Google Scholar

[7]

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

[8]

S. Aouaoui and F. S. B. Albuquerque, A weighted Trudinger-Moser type inequality and its applications to quasilinear elliptic problems with critical growth in the whole Euclidean space, Topol. Meth. Nonlinear Anal., 54 (2019), 109-130.  doi: 10.12775/tmna.2019.027.  Google Scholar

[9]

M. Calanchi, Some weighted inequalities of Trudinger-Moser Type. In: Analysis and Topology in Nonlinear Differential Equations, in Progress in Nonlinear Differential Equations and Applications, Springer, Birkhauser, Vol. 85, (2014), 163–174.  Google Scholar

[10]

M. Calanchi and B. Ruf, On Trudinger-Moser type inequalities with logarithmic weights, J. Differ. Equ., 258 (2015), 1967-1989.  doi: 10.1016/j.jde.2014.11.019.  Google Scholar

[11]

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.  Google Scholar

[12]

M. Calanchi, B. Ruf and F. Sani, Elliptic equations in dimension 2 with double exponential nonlinearities, NoDea Nonlinear Differ. Equ. Appl., 24 (2017), Art. 29. doi: 10.1007/s00030-017-0453-y.  Google Scholar

[13]

M. Calanchi and E. Terraneo, Non-radial maximizers for functionals with exponential nonlinearity in $\mathbb{R}^2$, Adv. Nonlinear Stud., 5 (2005), 337-350.  doi: 10.1515/ans-2005-0302.  Google Scholar

[14]

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

[15]

A. C. Cavalheiro, Weighted sobolev spaces and degenerate elliptic equations, Bol. Soc. Paran. Mat., 26 (2008), 117-132.  doi: 10.5269/bspm.v26i1-2.7415.  Google Scholar

[16]

J. F. De Oliveira and J.M. do Ò, Trudinger-Moser type inequalities for weighted spaces involving fractional dimensions, Proc. Amer. Math. Soc., 142 (2014), 2813-2828.  doi: 10.1090/S0002-9939-2014-12019-3.  Google Scholar

[17]

J. M. do Ò, N-Laplacian equations in $\mathbb{R}^N$ with critical growth, Abstr. Appl. Anal., 222 (1997), 301-315.  doi: 10.1155/S1085337597000419.  Google Scholar

[18]

J. M. do Ò and M. de Souza, On a class of singular Trudinger-Moser inequalities, Math. Nachr., 284 (2011), 1754-1776.  doi: 10.1002/mana.201000083.  Google Scholar

[19]

D. E. Edmunds, H. Hudzik and M. Krbec, On weighted critical imbeddings of Sobolev spaces, Math. Z., 286 (2011), 585-592.  doi: 10.1007/s00209-010-0684-7.  Google Scholar

[20]

M. F. Furtado, E. S. Medeiros and U. B. Severo, A Trudinger-Moser inequality in a weighted Sobolev space and applications, Math. Nach., 287 (2014), 1255-1273.  doi: 10.1002/mana.201200315.  Google Scholar

[21]

S. Goyal and K. Sreenadh, The Nehari manifold approach for $ N-$Laplace equation with singular and exponential nonlinearities in $ \mathbb{R}^2 $, Commun. Contemp. Math., (2014), Art. 1450011. doi: 10.1142/S0219199714500114.  Google Scholar

[22]

T. Kilpeläinen, Weighted Sobolev spaces and capacity, Ann. Acad. Sci. Fenn. Math., 19 (1994), 95-113.   Google Scholar

[23]

N. Lam, Sharp Trudinger-Moser inequalities with monomial weights, Nonlinear Differ. Equ. Appl., 24 (2017), Art. 39. doi: 10.1007/s00030-017-0456-8.  Google Scholar

[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.  Google Scholar

[25]

E. Nakai, N. Tomita and K. Yabuta, Density of the set of all infinitely differentiable functions with compact support in weighted Sobolev spaces, Sc. Math. Jpn., 10 (2004), 39-45.   Google Scholar

[26]

P. Pucci and V. Radulescu, The Impact of the Mountain Pass Theory in Nonlinear Analysis: a Mathematical Survey, Bollettino dell'Unione Matematica Italiana Serie, 9 (2010), 543-582.   Google Scholar

[27]

B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $\mathbb{R}^2$, J. Funct. Anal., 219 (2005), 340-367.  doi: 10.1016/j.jfa.2004.06.013.  Google Scholar

[28]

B. Ruf and F. Sani, Ground states for elliptic equations in $ \mathbb{R}^2 $ with exponential critical growth, in Geometric Properties for Parabolic and Elliptic PDE'S, Springer Serie, Vol. 2,251–268. doi: 10.1007/978-88-470-2841-8_16.  Google Scholar

[29]

W. A. Strauss, Existence of solitary waves in higher dimensions, Commun. Math. Phys., 55 (1977), 149-162.   Google Scholar

[30]

N. S. Trudinger, On the embedding into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-484.  doi: 10.1512/iumj.1968.17.17028.  Google Scholar

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