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arXiv · 2512.18562

Infinitely many solutions for a class of resonant problems

Philip Korman
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Abstract

We consider radially symmetric solutions for a class of resonant problems on a unit ball $B \subset R^n$ around the origin \[ \Delta u+\la _1 u +g(u)=f(r) \s \mbox{for $x \in B$}, \s u=0 \s \mbox{on $\partial B$} \,. \] Here the function $g(u)$ is periodic of mean zero, $x \in R^n$, $r=|x|$, $\la _1$ is the principal eigenvalue of $\Delta$ 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: $1 \leq n \leq 3$, $n=4$, $n=5$, $n=6$, and $n \geq 7$.

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Philip Korman. 2025-12-21. Infinitely many solutions for a class of resonant problems. https://arxiv.org/abs/2512.18562

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