Multiple Solutions for a Class of N -Laplacian Equations with Critical Growth and Indefinite Weight

Abstract

Using the suitable Trudinger-Moser inequality and the Mountain Pass Theorem, we prove the existence of multiple solutions for a class of $N$-Laplacian equations with critical growth and indefinite weight $-\text{div}\left({\left|\nabla u\right|}^{N-2}\nabla u\right)+V\left(x\right){\left|u\right|}^{N-2}u=\lambda \left({\left|u\right|}^{N-2}u/{\left|x\right|}^{\beta }\right)+\left(f\left(x,u\right)/{\left|x\right|}^{\beta }\right)+\mathrm{\varepsilon }h\left(x\right)$, $x\in {ℝ}^N$, $u\ne 0$, $x\in {ℝ}^N$, where , $V(x)$ is an indefinite weight, $f:{ℝ}^N\times ℝ\to ℝ$ behaves like $\text{exp}\left(\alpha {\left|u\right|}^{N/\left(N-1\right)}\right)$ and does not satisfy the Ambrosetti-Rabinowitz condition, and $h\in {(W^{\mathrm{1},N}({ℝ}^N))}^{\mathrm{*}}$.