Electronic Journal of Differential Equations,
Vol. 2012 (2012), No. 16, pp. 1-15.
Title: Multiple solutions for a q-Laplacian equation on an annulus
Authors: Shijian Tai (Shenzhen Experimental Education Group, China)
Jiangtao Wang (Zhongnan Univ. of Economics and Law, Wuhan, China)
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.