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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

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. $

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 and 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.

[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.

[3]

F. S. B. AlbuquerqueC. 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.

[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.

[5]

C. O. AlvesM. 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.

[6]

C. O. AlvesD. CassaniC. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[19]

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

[20]

M. F. FurtadoE. 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.

[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.

[22]

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

[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.

[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.

[25]

E. NakaiN. 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. 

[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. 

[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.

[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.

[29]

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

[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.

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.

[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.

[3]

F. S. B. AlbuquerqueC. 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.

[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.

[5]

C. O. AlvesM. 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.

[6]

C. O. AlvesD. CassaniC. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[19]

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

[20]

M. F. FurtadoE. 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.

[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.

[22]

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

[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.

[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.

[25]

E. NakaiN. 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. 

[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. 

[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.

[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.

[29]

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

[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.

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