Weigthed elliptic equation of Kirchhoff type with exponential non linear growth

Abstract

This work is concerned with the existence of a positive ground state solution for the following non local weighted problem
\begin{equation*}
\displaystyle \left\{
\begin{array}{rclll}
L_{(\sigma,V)}u &= & \displaystyle f(x,u)& \mbox{in} \ B \\
u &>&0 &\mbox{in }B\\
u&=&0 &\mbox{on } \partial B,
\end{array}
\right.
\end{equation*}
where $$L_{(\sigma,V)}u:=g(\int_{B}(\sigma(x)|\nabla u|^{N}+V(x)|u|^{N})dx)\big[-\textmd{div} (\sigma(x)|\nabla u|^{N-2} \nabla u)+V(x)|u|^{N-2}u\big],$$ B is the unit ball of $\mathbb{R}^{N}$, $ N>2$, $\sigma(x)=\Big(\log(\frac{e}{|x|})\Big)^{\beta(N-1)}$, $\beta \in[0,1)$ the singular logarithm weight , $V(x)$ is a positif continuous potential.The Kirchhoff function $g$ is positive and continuous on $(0,+\infty)$.
The nonlinearities are critical or subcritical growth in view of Trudinger-Moser inequalities of
exponential type. We prove the existence of a positive ground state solution by using Mountain Pass theorem .
In the critical case, the Euler-Lagrange function loses compactness except for a certain level. We dodge this problem by using adapted test functions to identify this level of compactness.