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

Abstract: 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 $$ \left{ \begin{array}{l} - \Delta_1 u +\xi \frac{u}{|u|} =\lambda |u|^{q-2}u+|u|^{1^*-2}u, \quad\text{in }\Omega, u=0, \quad\text{on } \partial\Omega. \end{array} \right. $$ 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 $\xi=1$, $\Omega = {x \in \mathbb{R}^N\,:\,r < |x| < r+1}$, $N\geq 2$, $N \not = 3$ 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.