### 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.

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.

#### Keywords

Variational methods; positive solutions; quasilinear equations

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