On a class of Kirchhoff type problem involving the 1-Laplacian and critical nonlinearity

Abstract

In this work, we study $1$-Laplacian Kirchhoff type equation with critical nonlinearity. Some of the main novelties here are the presence of the Kirchhoff function $M$ and the study of the asymptotic behavior of the solutions. We consider both the nondegenerate and the degenerate settings, namely, $M(0)> 0$ and $M(0)=0$, respectively. We show existence results and an asymptotic behavior of the solutions using an approximation method, in which the solutions are obtained as limit of solutions of a $p$-Kirchhoff type problems. To the best of our knowledge, our results are new even in local case $M\equiv1$. To overcome the lack of compactness, we use a version of the well-known Concentration Compactness Principle of Lions.