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 \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 $Δ$ 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: $1 \leq n \leq 3$, $n=4$, $n=5$, $n=6$, and $n \geq 7$.

💡 Research Summary

The paper investigates radially symmetric solutions of a resonant semilinear elliptic problem posed on the unit ball (B\subset\mathbb{R}^n): \