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
$$
\cases
-\Delta_{N} u = \lambda f(|x|,u) &x\in \Omega_r,\\
u > 0 &x\in \Omega_r,\\
u=0 &x\in \partial\Omega_r,
\endcases
\tag P
$$
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= \div(|\nabla u|^{N-2}\nabla u ) $
is the $N$-Laplacian operator and $f$ is a continuous function with
exponential critical growth.