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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 |
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. $
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. 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. |
| [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. 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. |
| [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. |
| [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. 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. |
| [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. |
| [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. 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.
|
| [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. 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. |
| [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. 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. |
| [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. |
| [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. 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. |
| [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. |
| [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. 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.
|
| [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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