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 $$\{\begin{array}{l}-{\Delta }_{1}u+\xi \frac{u}{|u|}=\lambda |u{|}^{q-2}u+|u{|}^{1^{\ast }-2}u,\text{in }\Omega ,u=0,\text{on }\partial \Omega .\end{array}$$ where $\Omega$ is a smooth bounded domain in ${\mathbb{R}}^N$, $N\ge 2$ and $\xi \in \{0,1\}$. Moreover, $\lambda >0$, $q\in (1,1^{\ast })$ and $1^{\ast }=\frac{N}{N-1}$. The first main result establishes the existence of many rotationally non-equivalent and nonradial solutions by assuming that $\xi =1$, $\Omega =\{x\in {\mathbb{R}}^N:r<|x|<r+1\}$, $N\ge 2$, $N\ne 3$ 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.