$ N- $Laplacian problems with critical double exponential nonlinearities

Shengbing Deng; Tingxi Hu; Chun-Lei Tang

doi:10.3934/dcds.2020306

Article Contents

2021, Volume 41, Issue 2: 987-1003. Doi: 10.3934/dcds.2020306

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$ N- $Laplacian problems with critical double exponential nonlinearities

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

^* Corresponding author: Chun-Lei Tang
^* Corresponding author: Chun-Lei Tang

Received: August 31, 2019

Revised: November 30, 2019

Early access: August 2020

Published: February 2021

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

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Abstract

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.

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References

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