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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  February 2021 Early access  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 and 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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.
[12]

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

[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.

[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.

[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.

[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.

[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.

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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.
[12]

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

[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.

[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.

[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.

[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.

[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.

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