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$ N- $Laplacian problems with critical double exponential nonlinearities

  • * Corresponding author: Chun-Lei Tang

    * Corresponding author: Chun-Lei Tang

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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  • 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 $.

    Mathematics Subject Classification: Primary:35J20, 35J25;Secondary:35J62.


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