Infinitely many solutions for a class of resonant problems

Abstract

We consider radially symmetric solutions for a class of resonant problems on a unit ball $B\subset R^n$ around the origin \[ Δu+\la _1 u +g(u)=f(r) \s \mbox{for $x\in B$}, \s u=0 \s \mbox{on $\partial B$} \,. \] Here the function $g(u)$ is periodic of mean zero, $x\in R^n$, $r=|x|$, $\la _1$ is the principal eigenvalue of $\hat{I}”$ 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 $n\ge 7$.

20 pages, 4 figures, comments are welcome