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doi: 10.3934/dcds.2021137
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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

*Corresponding author: Sami Aouaoui

Received  April 2021 Early access September 2021

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.

Citation: Sami Aouaoui, Rahma Jlel. Singular weighted sharp Trudinger-Moser inequalities defined on $ \mathbb{R}^N $ and applications to elliptic nonlinear equations. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021137
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[8]

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. Differential Equations, 261 (2016), 1933-1972.  doi: 10.1016/j.jde.2016.04.021.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 2011.  Google Scholar

[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.   Google Scholar

[15]

M. CalanchiE. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

D. G. de FigueiredoO. 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.  Google Scholar

[22]

S. DengT. 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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[26]

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

[27]

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

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

E. NakaiN. 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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[8]

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. Differential Equations, 261 (2016), 1933-1972.  doi: 10.1016/j.jde.2016.04.021.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 2011.  Google Scholar

[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.   Google Scholar

[15]

M. CalanchiE. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

D. G. de FigueiredoO. 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.  Google Scholar

[22]

S. DengT. 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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[26]

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

[27]

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

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

E. NakaiN. 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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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