Electron. J. Diff. Equ., Vol. 2012 (2012), No. 16, pp. 1-15.

Multiple solutions for a q-Laplacian equation on an annulus

Shijian Tai, Jiangtao Wang

Abstract:
In this article, we study the q-Laplacian equation
$$ -\Delta_{q}u=\big||x|-2\big|^{a}u^{p-1},\quad 1<|x|<3 , $$
where $\Delta_{q}u=\hbox{div}(|\nabla u|^{q-2} \nabla u)$ and $q>1$. We prove that the problem has two solutions when $a$ is large, and has two additional solutions when $p$ is close to the critical Sobolev exponent $q^{*}=\frac{Nq}{N-q}$. A symmetry-breaking phenomenon appears which shows that the least-energy solution cannot be radial function.

Submitted November 7, 2011. Published January 24, 2012.
Math Subject Classifications: 35J40.
Key Words: Ground state; minimizer; nonradial function; q-Laplacian; Rayleigh quotient.

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Shijian Tai
Shenzhen Experimental Education Group
Shenzhen, 5182028, China
email: 363846618@qq.com
Jiangtao Wang
School of Statistics and Mathematics
Zhongnan University of Economics and Law
Wuhan, 430073, China
email: wjtao1983@yahoo.com.cn

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