• Previous Article
    Global bifurcation for the Hénon problem
  • CPAA Home
  • This Issue
  • Next Article
    Uniqueness of positive radial solutions of the Brezis-Nirenberg problem on thin annular domains on $ {\mathbb S}^n $ and symmetry breaking bifurcations
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 & 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. 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.  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. 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.  Google Scholar

[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.  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. 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.  Google Scholar

[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.  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. 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.   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. 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.  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. 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.  Google Scholar

[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.  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. 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.  Google Scholar

[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.  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. 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.   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

[1]

Anouar Bahrouni. Trudinger-Moser type inequality and existence of solution for perturbed non-local elliptic operators with exponential nonlinearity. Communications on Pure & Applied Analysis, 2017, 16 (1) : 243-252. doi: 10.3934/cpaa.2017011

[2]

Tomasz Cieślak. Trudinger-Moser type inequality for radially symmetric functions in a ring and applications to Keller-Segel in a ring. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2505-2512. doi: 10.3934/dcdsb.2013.18.2505

[3]

Kyril Tintarev. Is the Trudinger-Moser nonlinearity a true critical nonlinearity?. Conference Publications, 2011, 2011 (Special) : 1378-1384. doi: 10.3934/proc.2011.2011.1378

[4]

Xiaobao Zhu. Remarks on singular trudinger-moser type inequalities. Communications on Pure & Applied Analysis, 2020, 19 (1) : 103-112. doi: 10.3934/cpaa.2020006

[5]

Guozhen Lu, Yunyan Yang. Sharp constant and extremal function for the improved Moser-Trudinger inequality involving $L^p$ norm in two dimension. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 963-979. doi: 10.3934/dcds.2009.25.963

[6]

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

[7]

Kanishka Perera, Marco Squassina. Bifurcation results for problems with fractional Trudinger-Moser nonlinearity. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 561-576. doi: 10.3934/dcdss.2018031

[8]

Van Hoang Nguyen. The Hardy–Moser–Trudinger inequality via the transplantation of Green functions. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3559-3574. doi: 10.3934/cpaa.2020155

[9]

Xumin Wang. Singular Hardy-Trudinger-Moser inequality and the existence of extremals on the unit disc. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2717-2733. doi: 10.3934/cpaa.2019121

[10]

Changliang Zhou, Chunqin Zhou. Extremal functions of Moser-Trudinger inequality involving Finsler-Laplacian. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2309-2328. doi: 10.3934/cpaa.2018110

[11]

Prosenjit Roy. On attainability of Moser-Trudinger inequality with logarithmic weights in higher dimensions. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5207-5222. doi: 10.3934/dcds.2019212

[12]

Yamin Wang. On nonexistence of extremals for the Trudinger-Moser functionals involving $ L^p $ norms. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4257-4268. doi: 10.3934/cpaa.2020191

[13]

Changliang Zhou, Chunqin Zhou. On the anisotropic Moser-Trudinger inequality for unbounded domains in $ \mathbb R^{n} $. Discrete & Continuous Dynamical Systems - A, 2020, 40 (2) : 847-881. doi: 10.3934/dcds.2020064

[14]

Nguyen Lam. Equivalence of sharp Trudinger-Moser-Adams Inequalities. Communications on Pure & Applied Analysis, 2017, 16 (3) : 973-998. doi: 10.3934/cpaa.2017047

[15]

Kenji Nakanishi, Tristan Roy. Global dynamics above the ground state for the energy-critical Schrödinger equation with radial data. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2023-2058. doi: 10.3934/cpaa.2016026

[16]

Chenmin Sun, Hua Wang, Xiaohua Yao, Jiqiang Zheng. Scattering below ground state of focusing fractional nonlinear Schrödinger equation with radial data. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2207-2228. doi: 10.3934/dcds.2018091

[17]

Jincai Kang, Chunlei Tang. Ground state radial sign-changing solutions for a gauged nonlinear Schrödinger equation involving critical growth. Communications on Pure & Applied Analysis, 2020, 19 (11) : 5239-5252. doi: 10.3934/cpaa.2020235

[18]

Yanjun Liu, Chungen Liu. Ground state solution and multiple solutions to elliptic equations with exponential growth and singular term. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2819-2838. doi: 10.3934/cpaa.2020123

[19]

Martin D. Buhmann, Slawomir Dinew. Limits of radial basis function interpolants. Communications on Pure & Applied Analysis, 2007, 6 (3) : 569-585. doi: 10.3934/cpaa.2007.6.569

[20]

Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020296

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (120)
  • HTML views (69)
  • Cited by (0)

Other articles
by authors

[Back to Top]