Infinitely many solutions for a class of resonant problems

TL;DR

This paper investigates the existence of solutions for a class of resonant boundary value problems with radially symmetric solutions, revealing that the number of solutions depends on the space dimension, with different behaviors for various dimensions.

Contribution

It provides a comprehensive analysis of the solution multiplicity for resonant problems on a unit ball, highlighting the dimension-dependent nature of solutions.

Findings

For 1 ≤ n ≤ 3, infinitely many solutions exist.

For n=4, the number of solutions is finite.

For n ≥ 7, solutions are finite or nonexistent depending on parameters.

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 $n \geq 7$.