doi: 10.3934/dcds.2020306

$ N- $Laplacian problems with critical double exponential nonlinearities

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 $
.
Citation: Shengbing Deng, Tingxi Hu, Chun-Lei Tang. $ N- $Laplacian problems with critical double exponential nonlinearities. Discrete & Continuous Dynamical Systems - A, 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

[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. 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

[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 FigueiredoO. 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

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. 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

[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 FigueiredoO. 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

[1]

Shiqiu Fu, Kanishka Perera. On a class of semipositone problems with singular Trudinger-Moser nonlinearities. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020452

[2]

Mathew Gluck. Classification of solutions to a system of $ n^{\rm th} $ order equations on $ \mathbb R^n $. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5413-5436. doi: 10.3934/cpaa.2020246

[3]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075

[4]

Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $ p $-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020445

[5]

Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $ p $ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020442

[6]

Lei Liu, Li Wu. Multiplicity of closed characteristics on $ P $-symmetric compact convex hypersurfaces in $ \mathbb{R}^{2n} $. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020378

[7]

Federico Rodriguez Hertz, Zhiren Wang. On $ \epsilon $-escaping trajectories in homogeneous spaces. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 329-357. doi: 10.3934/dcds.2020365

[8]

Aihua Fan, Jörg Schmeling, Weixiao Shen. $ L^\infty $-estimation of generalized Thue-Morse trigonometric polynomials and ergodic maximization. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 297-327. doi: 10.3934/dcds.2020363

[9]

Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020447

[10]

Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440

[11]

Wenjun Liu, Yukun Xiao, Xiaoqing Yue. Classification of finite irreducible conformal modules over Lie conformal algebra $ \mathcal{W}(a, b, r) $. Electronic Research Archive, , () : -. doi: 10.3934/era.2020123

[12]

Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077

[13]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138

[14]

Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020469

[15]

Peng Luo. Comparison theorem for diagonally quadratic BSDEs. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020374

[16]

Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296

[17]

Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253

[18]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217

[19]

Agnaldo José Ferrari, Tatiana Miguel Rodrigues de Souza. Rotated $ A_n $-lattice codes of full diversity. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020118

[20]

Adrian Constantin, Darren G. Crowdy, Vikas S. Krishnamurthy, Miles H. Wheeler. Stuart-type polar vortices on a rotating sphere. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 201-215. doi: 10.3934/dcds.2020263

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (61)
  • HTML views (134)
  • Cited by (0)

Other articles
by authors

[Back to Top]