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February  2021, 41(2): 987-1003. doi: 10.3934/dcds.2020306

$ N- $Laplacian problems with critical double exponential nonlinearities

Shengbing Deng , Tingxi Hu and  Chun-Lei Tang ,

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

* Corresponding author: Chun-Lei Tang

Received  September 2019 Revised  December 2019 Published  August 2020

Fund Project: The research was supported by National Nature Science Foundation of China (11971392, 11971393 and 11901473), Natural Science Foundation of Chongqing, China cstc2019jcyjjqX0022 and Fundamental Research Funds for the Central Universities XDJK2019TY001

In this paper, we prove the existence of a nontrivial solution for the following boundary value problem
$ \left\{ {\begin{array}{*{20}{l}}{ - {\rm{div}}(\omega (x)|\nabla u(x){|^{N - 2}}\nabla u(x)) = f(x,u),\;\;\quad }&{\;\;\;\;\;{\rm{in}}\;B;}\\{u = 0,\;\;\quad }&{\;\;\;\;\;{\rm{on}}\;\partial B,}\end{array}} \right.{\rm{ }}\;\;\;\;\;\;\;\left( 1 \right)$
where
$ B $
 is the unit ball in
$ \mathbb{R}^N $
,
$ N\geq 2 $
, the radial positive weight
$ \omega(x) $
 is of logarithmic type, the function
$ f(x,u) $
 is continuous in
$ B\times\mathbb{R} $
 and has critical double exponential growth, which behaves like
$ \exp\{e^{\alpha |u|^{\frac{N}{N-1}}}\} $
 as
$ |u|\to\infty $
 for some
$ \alpha>0 $
.
Keywords: $ N- $Laplacian, Trudinger-Moser type inequality, critical double exponential nonlinearities, mountain pass theorem.
Mathematics Subject Classification: Primary:35J20, 35J25;Secondary:35J62.
Citation: Shengbing Deng, Tingxi Hu, Chun-Lei Tang. $ N- $Laplacian problems with critical double exponential nonlinearities. Discrete & Continuous Dynamical Systems, 2021, 41 (2) : 987-1003. doi: 10.3934/dcds.2020306
References:
[1]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

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L. Boccardo and F. Murat, Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlinear Anal., 19 (1992), 581-597.  doi: 10.1016/0362-546X(92)90023-8.  Google Scholar

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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.  Google Scholar

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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

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M. Calanchi, B. Ruf and F. Sani, Elliptic equations in dimension 2 with double exponential nonlinearities, NoDEA Nonlinear Differential Equations Appl., 24 (2017), 29, 18 pp. doi: 10.1007/s00030-017-0453-y.  Google Scholar

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D. G. de Figueiredo, O. H. Miyagaki and B. Ruf, Elliptic equations in $R^2$ with nonlinearities in the critical growth range, Calc. Var. Partial Differential Equations, 3 (1995), 139-153.  doi: 10.1007/BF01205003.  Google Scholar

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J. M. B. do Ó, Semilinear Dirichlet problems for the $N$-Laplacian in $\bf R^N$ with nonlinearities in the critical growth range, Differential Integral Equations, 9 (1996), 967-979.   Google Scholar

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A. Kufner, Weighted Sobolev Spaces, Wiley, Hoboken, 1985. Google Scholar

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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

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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

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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

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N. S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-483.  doi: 10.1512/iumj.1968.17.17028.  Google Scholar

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M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

show all references

References:
[1]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[2]

L. Boccardo and F. Murat, Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlinear Anal., 19 (1992), 581-597.  doi: 10.1016/0362-546X(92)90023-8.  Google Scholar

[3]

M. Calanchi, Some weighted inequalities of Trudinger-Moser type, in: Progress in Nonlinear Differential Equations and Appl., Birkhäuser, 85 (2014), 163–174.  Google Scholar

[4]

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.  Google Scholar

[5]

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

[6]

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

[7]

M. Calanchi, B. Ruf and F. Sani, Elliptic equations in dimension 2 with double exponential nonlinearities, NoDEA Nonlinear Differential Equations Appl., 24 (2017), 29, 18 pp. doi: 10.1007/s00030-017-0453-y.  Google Scholar

[8]

D. G. de Figueiredo, O. H. Miyagaki and B. Ruf, Elliptic equations in $R^2$ with nonlinearities in the critical growth range, Calc. Var. Partial Differential Equations, 3 (1995), 139-153.  doi: 10.1007/BF01205003.  Google Scholar

[9]

S. Deng, Existence of solutions for some weighted mean field equations in dimension $N$, Appl. Math. Lett., 100 (2020), 106010, 7 pp. doi: 10.1016/j.aml.2019.106010.  Google Scholar

[10]

J. M. B. do Ó, Semilinear Dirichlet problems for the $N$-Laplacian in $\bf R^N$ with nonlinearities in the critical growth range, Differential Integral Equations, 9 (1996), 967-979.   Google Scholar

[11] Y. Jabri, The Mountain Pass Theorem, Variant, Generalizations and some Applications, Encyclopedia of Mathematics and its Applications, 95, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511546655.  Google Scholar
[12]

A. Kufner, Weighted Sobolev Spaces, Wiley, Hoboken, 1985. Google Scholar

[13]

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

[14]

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

[15]

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

[16]

N. S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-483.  doi: 10.1512/iumj.1968.17.17028.  Google Scholar

[17]

M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

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