Multiple solutions for a q-Laplacian equation on an annulus

In this article, we study the q-Laplacian equation $$ -Delta_{q}u=ig||x|-2ig|^{a}u^{p-1},quad 1<|x|<3 , $$ where $Delta_{q}u=hbox{div}(| abla u|^{q-2} abla u)$ 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.