Weighted Schr\"{o}dinger- Kirchhoff type problem in dimension 2 with non-linear double exponential growth

Abstract

\begin{abstract}
In this work, we study the weighted Kirchhoff problem
\begin{equation*}
\displaystyle \left\{
\begin{array}{rclll}
g\big(\int_{B}(\sigma(x)|\nabla u|^{2}+V(x)u^{2})dx\big)\big[-\textmd{div} (\sigma(x) \nabla u)+V(x)u \big] &=& \ f(x,u)& \mbox{in} & B \\
u &>&0 &\mbox{in }& B\\
u&=&0 &\mbox{on }& \partial B ,
\end{array}
\right.
\end{equation*}
where $B$ is the unit ball of $\mathbb{R}^{2}$, $ \sigma(x)=\log \frac{e}{|x|}$,
the singular logarithm weight in the Trudinger-Moser embedding, $g$ is a continuous positive function on $\mathbb{R^{+}}$ and the potential $V$ is a continuous positve function.
The nonlinearities are critical or subcritical growth in view of Trudinger-Moser
inequalities. We prove the existence of non-trivial solutions
via the critical point theory. In the critical case, the associated energy function does not satisfy
the condition of compactness. We provide a new condition for growth and we stress its importance
to check the min-max compactness level.

\end{abstract}