# American Institute of Mathematical Sciences

February  2021, 41(2): 987-1003. 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
 $\mathbb{R}^N$
,
 $N\geq 2$
 $\omega(x)$
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, 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 [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

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
 [1] Shiqiu Fu, Kanishka Perera. On a class of semipositone problems with singular Trudinger-Moser nonlinearities. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1747-1756. doi: 10.3934/dcdss.2020452 [2] 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 [3] Jong Yoon Hyun, Yoonjin Lee, Yansheng Wu. Connection of $p$-ary $t$-weight linear codes to Ramanujan Cayley graphs with $t+1$ eigenvalues. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020133 [4] Chaoqian Li, Yajun Liu, Yaotang Li. Note on $Z$-eigenvalue inclusion theorems for tensors. Journal of Industrial & Management Optimization, 2021, 17 (2) : 687-693. doi: 10.3934/jimo.2019129 [5] Ian Schindler, Kyril Tintarev. Mountain pass solutions to semilinear problems with critical nonlinearity. Conference Publications, 2007, 2007 (Special) : 912-919. doi: 10.3934/proc.2007.2007.912 [6] Shihu Li, Wei Liu, Yingchao Xie. Large deviations for stochastic 3D Leray-$\alpha$ model with fractional dissipation. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2491-2509. doi: 10.3934/cpaa.2019113 [7] Emma D'Aniello, Saber Elaydi. The structure of $\omega$-limit sets of asymptotically non-autonomous discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 903-915. doi: 10.3934/dcdsb.2019195 [8] Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $\mathbb{R}^2$. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1631-1648. doi: 10.3934/dcdss.2020447 [9] Raj Kumar, Maheshanand Bhaintwal. Duadic codes over $\mathbb{Z}_4+u\mathbb{Z}_4$. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020135 [10] Mingxin Wang, Qianying Zhang. Dynamics for the diffusive Leslie-Gower model with double free boundaries. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2591-2607. doi: 10.3934/dcds.2018109 [11] Manoel J. Dos Santos, Baowei Feng, Dilberto S. Almeida Júnior, Mauro L. Santos. Global and exponential attractors for a nonlinear porous elastic system with delay term. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2805-2828. doi: 10.3934/dcdsb.2020206 [12] Ka Luen Cheung, Man Chun Leung. Asymptotic behavior of positive solutions of the equation $\Delta u + K u^{\frac{n+2}{n-2}} = 0$ in $IR^n$ and positive scalar curvature. Conference Publications, 2001, 2001 (Special) : 109-120. doi: 10.3934/proc.2001.2001.109 [13] Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261 [14] Martial Agueh, Reinhard Illner, Ashlin Richardson. Analysis and simulations of a refined flocking and swarming model of Cucker-Smale type. Kinetic & Related Models, 2011, 4 (1) : 1-16. doi: 10.3934/krm.2011.4.1 [15] Andrea Cianchi, Adele Ferone. Improving sharp Sobolev type inequalities by optimal remainder gradient norms. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1363-1386. doi: 10.3934/cpaa.2012.11.1363 [16] Yahui Niu. A Hopf type lemma and the symmetry of solutions for a class of Kirchhoff equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021027 [17] Fernando P. da Costa, João T. Pinto, Rafael Sasportes. On the convergence to critical scaling profiles in submonolayer deposition models. Kinetic & Related Models, 2018, 11 (6) : 1359-1376. doi: 10.3934/krm.2018053 [18] Gioconda Moscariello, Antonia Passarelli di Napoli, Carlo Sbordone. Planar ACL-homeomorphisms : Critical points of their components. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1391-1397. doi: 10.3934/cpaa.2010.9.1391 [19] A. Aghajani, S. F. Mottaghi. Regularity of extremal solutions of semilinaer fourth-order elliptic problems with general nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (3) : 887-898. doi: 10.3934/cpaa.2018044 [20] Changpin Li, Zhiqiang Li. Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021023

2019 Impact Factor: 1.338