Multiplicity of solutions for a class of quasilinear problems involving the $1$-Laplacian operator with critical growth

The aim of this paper is to establish two results about multiplicity of solutions to problems involving the $1-$Laplacian operator, with nonlinearities with critical growth. To be more specific, we study the following problem $$ left{ begin{array}{l} - Delta_1 u +xi frac{u}{|u|} =lambda |u|^{q-2}u+|u|^{1^*-2}u, quadtext{in }Omega, u=0, quadtext{on } partialOmega. end{array} right. $$ where $Omega$ is a smooth bounded domain in $mathbb{R}^N$, $N geq 2$ and $xi in{0,1}$. Moreover, $lambda > 0$, $q in (1,1^*)$ and $1^*=frac{N}{N-1}$. The first main result establishes the existence of many rotationally non-equivalent and nonradial solutions by assuming that $xi=1$, $Omega = {x in mathbb{R}^N,:,r < |x| < r+1}$, $Ngeq 2$, $N ot = 3$ and $r > 0$. In the second one, $Omega$ is a smooth bounded domain, $xi=0$, and the multiplicity of solutions is proved through an abstract result which involves genus theory for functionals which are sum of a $C^1$ functional with a convex lower semicontinuous functional.