Infinitely many solutions for a class of resonant problems

We consider radially symmetric solutions for a class of resonant problems on a unit ball $B\subset R^n$ around the origin Δu+\la _1 u +g(u)=f(r) \s for $x \in B$, \s u=0 \s on $\partial B$ \,. Here the function $g(u)$ is periodic of mean zero, $x\in R^n$, $r=\mathrm{\mid }x\mathrm{\mid }$, ${\text{la}}_1$ is the principal eigenvalue of $\mathrm{\Delta }$ on $B$. The problem has either infinitely many or finitely many solutions depending on the space dimension $n$. The situation turns out to be different for each of the following cases: $1\le n\le 3$, $n=4$, $n=5$, $n=6$, and $n\ge 7$.