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Singular weighted sharp Trudinger-Moser inequalities defined on $ \mathbb{R}^N $ and applications to elliptic nonlinear equations
1. | University of Kairouan, High Institute of Applied Mathematics, and Informatics of Kairouan, , Avenue Assad Iben Fourat, Kairouan, 3100, Tunisia |
2. | University of Monastir, Faculty of Sciences of Monastir, Avenue de l'environnement 5019 Monastir, Tunisia |
This work comes to complete some previous ones of ours. Actually, in this paper, we establish some singular weighted inequalities of Trudinger-Moser type for radial functions defined on the whole euclidean space $ \mathbb{R}^N,\ N \geq 2. $ The weights considered are of logarithmic type. The singularity plays a capital role to prove the sharpness of the inequalities. These inequalities are later improved using some concentration-compactness arguments. The last part in this work is devoted to the application of the inequalities established to some singular elliptic nonlinear equations involving a new growth conditions at infinity of exponential type.
References:
[1] |
E. Abreu and L. G. Fernandez Jr,
On a weighted Trudinger-Moser inequality in $ \mathbb{R}^N$, J. Differential Equations, 269 (2020), 3089-3118.
doi: 10.1016/j.jde.2020.02.023. |
[2] |
S. Adachi and K. Tanaka,
Trudinger type inequalities in $ \mathbb{R}^N $ and their best exponents, Proc. Amer. Math. Soc., 128 (2000), 2051-2057.
doi: 10.1090/S0002-9939-99-05180-1. |
[3] |
Adimurthi and K. Sandeep, A singular Moser-Trudinger embedding and its applications, NoDEA Nonlinear Differential Equations Appl., 13 (2007) 585–603.
doi: 10.1007/s00030-006-4025-9. |
[4] |
Ad imurthi 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. |
[5] |
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. |
[6] |
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. |
[7] |
F. S. B. Albuquerque and S. Aouaoui,
A weighted Trudinger-Moser type inequality and its applications to quasilinear elliptic problems with critical growth in the whole Euclidean space, Topol. Methods Nonlinear Anal., 54 (2019), 109-130.
doi: 10.12775/tmna.2019.027. |
[8] |
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. Differential Equations, 261 (2016), 1933-1972.
doi: 10.1016/j.jde.2016.04.021. |
[9] |
S. Aouaoui,
A new Trudinger-Moser type inequality and an application to some elliptic equation with doubly exponential nonlinearity in the whole space $\mathbb{R}^2$, Arch. Math., 114 (2020), 199-214.
doi: 10.1007/s00013-019-01386-7. |
[10] |
S. Aouaoui and R. Jlel,
A new singular Trudinger-Moser type inequality with logarithmic weights and applications, Adv. Nonlinear Stud., 20 (2020), 113-139.
doi: 10.1515/ans-2019-2068. |
[11] |
S. Aouaoui and R. Jlel,
On some elliptic equation in the whole euclidean space $ \mathbb{R}^2 $ with nonlinearities having new exponential growth condition, Commun. Pure Appl. Anal., 19 (2020), 4771-4796.
doi: 10.3934/cpaa.2020211. |
[12] |
S. Aouaoui and R. Jlel, New weighted sharp Trudinger-Moser inequalities defined on the whole euclidean space $ \mathbb{R}^N $ and applications, Calc. Var. Partial Differential Equations, 60 (2021), 40pp.
doi: 10.1007/s00526-021-01925-7. |
[13] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 2011. |
[14] |
M. Calanchi,
Some weighted inequalities of Trudinger-Moser Type, In:, Analysis and Topology in Nonlinear Differential Equations, Nonlinear Differential Equations Appl., 85 (2014), 163-174.
|
[15] |
M. Calanchi, E. Massa and B. Ruf,
Weighted Trudinger-Moser inequalities and associated Liouville type equations, Proc. Amer. Math. Soc., 146 (2018), 5243-5256.
doi: 10.1090/proc/14189. |
[16] |
M. Calanchi and B. Ruf,
On Trudinger-Moser type inequalities with logarithmic weights, J. Differential Equations, 258 (2015), 1967-1989.
doi: 10.1016/j.jde.2014.11.019. |
[17] |
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. |
[18] |
M. Calanchi, B. Ruf and F. Sani, Elliptic equations in dimension 2 with double exponential nonlinearities, NoDEA Nonlinear Differential Equations Appl., 24 (2017), 18pp.
doi: 10.1007/s00030-017-0453-y. |
[19] |
D. M. Cao,
Nontrivial solution of semilinear elliptic equation with critical exponent in $\mathbb{R}^2$, Comm. Partial Differential Equations, 17 (1992), 407-435.
doi: 10.1080/03605309208820848. |
[20] |
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. |
[21] |
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. Partial Differential Equations, 3 (1995), 139-153.
doi: 10.1007/BF01205003. |
[22] |
S. Deng, T. Hu and C-L. Tang,
$ N-$Laplacian problems with critical double exponential nonlinearities, Discrete Contin. Dyn. Syst., 41 (2021), 987-1003.
doi: 10.3934/dcds.2020306. |
[23] |
J. F. de Oliveira and J. M. do Ò,
Trudinger-Moser type inequalities for weighted Sobolev spaces involving fractional dimensions, Proc. Amer. Math. Soc., 142 (2014), 2813-2828.
doi: 10.1090/S0002-9939-2014-12019-3. |
[24] |
J. M. do Ó,
Semilinear Dirichlet problems for the $n-$Laplacian in $ \mathbb{R}^n $ with nonlinearities in critical growth range, Differential Integral Equations, 9 (1996), 967-979.
|
[25] |
J. M. do Ò and M. de Souza,
On a class of singular Trudinger-Moser type inequalities and its applications, Math. Nachr., 284 (2011), 1754-1776.
doi: 10.1002/mana.201000083. |
[26] |
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. |
[27] |
T. Kilpeläinen,
Weighted Sobolev spaces and capacity, Ann. Acad. Sci. Fenn. Math., 19 (1994), 95-113.
|
[28] |
N. Lam, Sharp Trudinger-Moser inequalities with monomial weights, NoDEA Nonlinear Differ. Equ. Appl., 24 (2017).
doi: 10.1007/s00030-017-0456-8. |
[29] |
N. Lam and G. Lu,
Existence and multiplicity of solutions to equations of $n-$Laplacian type with critical exponential growth in $ \mathbb{R}^n$, J. Funct. Anal., 262 (2012), 1132-1165.
doi: 10.1016/j.jfa.2011.10.012. |
[30] |
X. Li,
An improved singular Trudinger-Moser inequality in $ \mathbb{R}^N $ and its extremal functions, J. Math. Anal. Appl., 462 (2018), 1109-1129.
doi: 10.1016/j.jmaa.2018.01.080. |
[31] |
Y. Li and B. Ruf,
A sharp Trudinger-Moser type inequality for unbounded domains in $ \mathbb{R}^n$, Indiana Univ. Math. J., 57 (2008), 451-480.
doi: 10.1512/iumj.2008.57.3137. |
[32] |
X. Li and Y. Yang, Extremal functions for singular Trudinger-Moser inequalities in the entire Euclidean space, J. Differential Equations, 264 (2018) 4901–4943.
doi: 10.1016/j.jde.2017.12.028. |
[33] |
P. L. Lions,
The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoamericana, 1 (1985), 145-201.
doi: 10.4171/RMI/6. |
[34] |
J. Moser,
A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20 (1970/71), 1077-1092.
doi: 10.1512/iumj.1971.20.20101. |
[35] |
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. Jpna., 60 (2004), 121-127.
|
[36] |
V. H. Nguyen and F. Takahashi,
On a weighted Trudinger-Moser type inequality on the whole space and related maximizing problem, Differential Integral Equations, 31 (2018), 785-806.
|
[37] |
V. H. Nguyen,
Remarks on the Moser-Trudinger type inequality with logarithmic weights in dimension N, Proc. Amer. Math. Soc., 147 (2019), 5183-5193.
doi: 10.1090/proc/14566. |
[38] |
P. Pucci and V. Radulescu,
The impact of the mountain pass theory in nonlinear analysis: A mathematical survey, Boll. Unione Mat. Ital., 3 (2010), 543-582.
|
[39] |
P. Roy,
Extremal function for Moser-Trudinger type inequality with logarithmic weight, Nonlinear Anal., 135 (2016), 194-204.
doi: 10.1016/j.na.2016.01.024. |
[40] |
P. Roy,
On attainability of Moser Trudinger inequality with logarithmic weights in higher dimensions, Discrete Contin. Dyn. Syst., 39 (2019), 5207-5222.
doi: 10.3934/dcds.2019212. |
[41] |
B. Ruf and F. Sani, Ground states for elliptic equations in $ \mathbb{R}^2 $ with exponential critical growth, Geometric properties for parabolic and elliptic PDE'S, Springer, Milan, 2 (2013), 251–268.
doi: 10.1007/978-88-470-2841-8_16. |
[42] |
N. S. Trudinger,
On embedding into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-484.
doi: 10.1512/iumj.1968.17.17028. |
[43] |
C. Zhang,
Concentration-Compactness principle for Trudinger-Moser inequalities with logarithmic weights and their applications, Nonlinear Anal., 197 (2020), 111845.
doi: 10.1016/j.na.2020.111845. |
show all references
References:
[1] |
E. Abreu and L. G. Fernandez Jr,
On a weighted Trudinger-Moser inequality in $ \mathbb{R}^N$, J. Differential Equations, 269 (2020), 3089-3118.
doi: 10.1016/j.jde.2020.02.023. |
[2] |
S. Adachi and K. Tanaka,
Trudinger type inequalities in $ \mathbb{R}^N $ and their best exponents, Proc. Amer. Math. Soc., 128 (2000), 2051-2057.
doi: 10.1090/S0002-9939-99-05180-1. |
[3] |
Adimurthi and K. Sandeep, A singular Moser-Trudinger embedding and its applications, NoDEA Nonlinear Differential Equations Appl., 13 (2007) 585–603.
doi: 10.1007/s00030-006-4025-9. |
[4] |
Ad imurthi 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. |
[5] |
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. |
[6] |
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. |
[7] |
F. S. B. Albuquerque and S. Aouaoui,
A weighted Trudinger-Moser type inequality and its applications to quasilinear elliptic problems with critical growth in the whole Euclidean space, Topol. Methods Nonlinear Anal., 54 (2019), 109-130.
doi: 10.12775/tmna.2019.027. |
[8] |
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. Differential Equations, 261 (2016), 1933-1972.
doi: 10.1016/j.jde.2016.04.021. |
[9] |
S. Aouaoui,
A new Trudinger-Moser type inequality and an application to some elliptic equation with doubly exponential nonlinearity in the whole space $\mathbb{R}^2$, Arch. Math., 114 (2020), 199-214.
doi: 10.1007/s00013-019-01386-7. |
[10] |
S. Aouaoui and R. Jlel,
A new singular Trudinger-Moser type inequality with logarithmic weights and applications, Adv. Nonlinear Stud., 20 (2020), 113-139.
doi: 10.1515/ans-2019-2068. |
[11] |
S. Aouaoui and R. Jlel,
On some elliptic equation in the whole euclidean space $ \mathbb{R}^2 $ with nonlinearities having new exponential growth condition, Commun. Pure Appl. Anal., 19 (2020), 4771-4796.
doi: 10.3934/cpaa.2020211. |
[12] |
S. Aouaoui and R. Jlel, New weighted sharp Trudinger-Moser inequalities defined on the whole euclidean space $ \mathbb{R}^N $ and applications, Calc. Var. Partial Differential Equations, 60 (2021), 40pp.
doi: 10.1007/s00526-021-01925-7. |
[13] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 2011. |
[14] |
M. Calanchi,
Some weighted inequalities of Trudinger-Moser Type, In:, Analysis and Topology in Nonlinear Differential Equations, Nonlinear Differential Equations Appl., 85 (2014), 163-174.
|
[15] |
M. Calanchi, E. Massa and B. Ruf,
Weighted Trudinger-Moser inequalities and associated Liouville type equations, Proc. Amer. Math. Soc., 146 (2018), 5243-5256.
doi: 10.1090/proc/14189. |
[16] |
M. Calanchi and B. Ruf,
On Trudinger-Moser type inequalities with logarithmic weights, J. Differential Equations, 258 (2015), 1967-1989.
doi: 10.1016/j.jde.2014.11.019. |
[17] |
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. |
[18] |
M. Calanchi, B. Ruf and F. Sani, Elliptic equations in dimension 2 with double exponential nonlinearities, NoDEA Nonlinear Differential Equations Appl., 24 (2017), 18pp.
doi: 10.1007/s00030-017-0453-y. |
[19] |
D. M. Cao,
Nontrivial solution of semilinear elliptic equation with critical exponent in $\mathbb{R}^2$, Comm. Partial Differential Equations, 17 (1992), 407-435.
doi: 10.1080/03605309208820848. |
[20] |
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. |
[21] |
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. Partial Differential Equations, 3 (1995), 139-153.
doi: 10.1007/BF01205003. |
[22] |
S. Deng, T. Hu and C-L. Tang,
$ N-$Laplacian problems with critical double exponential nonlinearities, Discrete Contin. Dyn. Syst., 41 (2021), 987-1003.
doi: 10.3934/dcds.2020306. |
[23] |
J. F. de Oliveira and J. M. do Ò,
Trudinger-Moser type inequalities for weighted Sobolev spaces involving fractional dimensions, Proc. Amer. Math. Soc., 142 (2014), 2813-2828.
doi: 10.1090/S0002-9939-2014-12019-3. |
[24] |
J. M. do Ó,
Semilinear Dirichlet problems for the $n-$Laplacian in $ \mathbb{R}^n $ with nonlinearities in critical growth range, Differential Integral Equations, 9 (1996), 967-979.
|
[25] |
J. M. do Ò and M. de Souza,
On a class of singular Trudinger-Moser type inequalities and its applications, Math. Nachr., 284 (2011), 1754-1776.
doi: 10.1002/mana.201000083. |
[26] |
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. |
[27] |
T. Kilpeläinen,
Weighted Sobolev spaces and capacity, Ann. Acad. Sci. Fenn. Math., 19 (1994), 95-113.
|
[28] |
N. Lam, Sharp Trudinger-Moser inequalities with monomial weights, NoDEA Nonlinear Differ. Equ. Appl., 24 (2017).
doi: 10.1007/s00030-017-0456-8. |
[29] |
N. Lam and G. Lu,
Existence and multiplicity of solutions to equations of $n-$Laplacian type with critical exponential growth in $ \mathbb{R}^n$, J. Funct. Anal., 262 (2012), 1132-1165.
doi: 10.1016/j.jfa.2011.10.012. |
[30] |
X. Li,
An improved singular Trudinger-Moser inequality in $ \mathbb{R}^N $ and its extremal functions, J. Math. Anal. Appl., 462 (2018), 1109-1129.
doi: 10.1016/j.jmaa.2018.01.080. |
[31] |
Y. Li and B. Ruf,
A sharp Trudinger-Moser type inequality for unbounded domains in $ \mathbb{R}^n$, Indiana Univ. Math. J., 57 (2008), 451-480.
doi: 10.1512/iumj.2008.57.3137. |
[32] |
X. Li and Y. Yang, Extremal functions for singular Trudinger-Moser inequalities in the entire Euclidean space, J. Differential Equations, 264 (2018) 4901–4943.
doi: 10.1016/j.jde.2017.12.028. |
[33] |
P. L. Lions,
The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoamericana, 1 (1985), 145-201.
doi: 10.4171/RMI/6. |
[34] |
J. Moser,
A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20 (1970/71), 1077-1092.
doi: 10.1512/iumj.1971.20.20101. |
[35] |
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. Jpna., 60 (2004), 121-127.
|
[36] |
V. H. Nguyen and F. Takahashi,
On a weighted Trudinger-Moser type inequality on the whole space and related maximizing problem, Differential Integral Equations, 31 (2018), 785-806.
|
[37] |
V. H. Nguyen,
Remarks on the Moser-Trudinger type inequality with logarithmic weights in dimension N, Proc. Amer. Math. Soc., 147 (2019), 5183-5193.
doi: 10.1090/proc/14566. |
[38] |
P. Pucci and V. Radulescu,
The impact of the mountain pass theory in nonlinear analysis: A mathematical survey, Boll. Unione Mat. Ital., 3 (2010), 543-582.
|
[39] |
P. Roy,
Extremal function for Moser-Trudinger type inequality with logarithmic weight, Nonlinear Anal., 135 (2016), 194-204.
doi: 10.1016/j.na.2016.01.024. |
[40] |
P. Roy,
On attainability of Moser Trudinger inequality with logarithmic weights in higher dimensions, Discrete Contin. Dyn. Syst., 39 (2019), 5207-5222.
doi: 10.3934/dcds.2019212. |
[41] |
B. Ruf and F. Sani, Ground states for elliptic equations in $ \mathbb{R}^2 $ with exponential critical growth, Geometric properties for parabolic and elliptic PDE'S, Springer, Milan, 2 (2013), 251–268.
doi: 10.1007/978-88-470-2841-8_16. |
[42] |
N. S. Trudinger,
On embedding into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-484.
doi: 10.1512/iumj.1968.17.17028. |
[43] |
C. Zhang,
Concentration-Compactness principle for Trudinger-Moser inequalities with logarithmic weights and their applications, Nonlinear Anal., 197 (2020), 111845.
doi: 10.1016/j.na.2020.111845. |
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