<inline-formula><tex-math id="M1">$ N- $</tex-math></inline-formula>Laplacian problems with critical double exponential nonlinearities
Shengbing Deng, Tingxi Hu, Chun‐Lei Tang
Abstract
<p style='text-indent:20px;'>In this paper, we prove the existence of a nontrivial solution for the following boundary value problem <p style='text-indent:20px;'><disp-formula> <label>1</label> <tex-math id="E1"> \begin{document}$ \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)$\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>where <inline-formula><tex-math id="M2">\begin{document}$ B $\end{document}</tex-math></inline-formula> is the unit ball in <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{R}^N $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ N\geq 2 $\end{document}</tex-math></inline-formula>, the radial positive weight <inline-formula><tex-math id="M5">\begin{document}$ \omega(x) $\end{document}</tex-math></inline-formula> is of logarithmic type, the function <inline-formula><tex-math id="M6">\begin{document}$ f(x,u) $\end{document}</tex-math></inline-formula> is continuous in <inline-formula><tex-math id="M7">\begin{document}$ B\times\mathbb{R} $\end{document}</tex-math></inline-formula> and has critical double exponential growth, which behaves like <inline-formula><tex-math id="M8">\begin{document}$ \exp\{e^{\alpha |u|^{\frac{N}{N-1}}}\} $\end{document}</tex-math></inline-formula> as <inline-formula><tex-math id="M9">\begin{document}$ |u|\to\infty $\end{document}</tex-math></inline-formula> for some <inline-formula><tex-math id="M10">\begin{document}$ \alpha>0 $\end{document}</tex-math></inline-formula>.