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

TL;DR

This paper proves the existence of multiple solutions, including nonradial and non-equivalent solutions, for a class of quasilinear problems involving the 1-Laplacian operator with critical growth, using variational methods and genus theory.

Contribution

It establishes new multiplicity results for 1-Laplacian problems with critical growth, including nonradial solutions and solutions in general domains, using advanced variational techniques.

Findings

Existence of many nonradial solutions in specific annular domains.

Multiplicity of solutions in general domains via genus theory.

Results extend understanding of 1-Laplacian problems with critical nonlinearities.

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.