## Abstract and Applied Analysis

### 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 $-\text{div}({|\nabla u|}^{N-2}\nabla u)+V(x){|u|}^{N-2}u=\lambda ({|u|}^{N-2}u/{|x|}^{\beta })+(f(x,u)/{|x|}^{\beta })+\mathrm{\varepsilon }h(x)$, $x\in {\Bbb R}^{N}$, $u\ne 0$, $x\in {\Bbb R}^{N}$, where $\mathrm{0}<\beta , $V(x)$ is an indefinite weight, $f:{\Bbb R}^{N}{\times}\Bbb R\to \Bbb R$ behaves like $\text{exp}(\alpha {|u|}^{N/(N-1)})$ and does not satisfy the Ambrosetti-Rabinowitz condition, and $h\in {({W}^{\mathrm{1},N}({\Bbb R}^{N}))}^{\mathrm{\ast}}$.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 942092, 14 pages.

Dates
First available in Project Euclid: 26 March 2014

https://projecteuclid.org/euclid.aaa/1395858341

Digital Object Identifier
doi:10.1155/2014/942092

Mathematical Reviews number (MathSciNet)
MR3166669

#### Citation

Zhang, Guoqing; Yao, Ziyan. Multiple Solutions for a Class of $N$ -Laplacian Equations with Critical Growth and Indefinite Weight. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 942092, 14 pages. doi:10.1155/2014/942092. https://projecteuclid.org/euclid.aaa/1395858341

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