Multiplicity of nonradial solutions for a class of quasilinear equations on annulus with exponential critical growth

Abstract

In this paper, we establish the existence of many rotationally non-equivalent and nonradial solutions for the following class of quasilinear problems \begin{equation} \begin{cases} -\Delta_{N} u = \lambda f(|x|,u) &x\in \Omega_r,\\ u > 0 &x\in \Omega_r,\\ u=0 &x\in \partial\Omega_r, \end{cases} \tag{$\textrm{P}$} \end{equation} where $\Omega_r = \{ x \in \mathbb{R}^{N}: r < |x| < r+1\}$, $N \geq 2$, $N\neq 3$, $r > 0$, $\lambda > 0$, $\Delta_{N}u= {\rm div}(|\nabla u|^{N-2}\nabla u ) $ is the $N$-Laplacian operator and $f$ is a continuous function with exponential critical growth.