Instability of solutions to Kirchhoff type problems in low dimension
Volume 124 / 2020
Annales Polonici Mathematici 124 (2020), 75-91
MSC: 35J92, 35J25, 35B53, 35B35.
DOI: 10.4064/ap181120-3-5
Published online: 21 November 2019
Abstract
We study the Kirchhoff type problem \[ \begin {cases} - m\Bigl (\displaystyle \int _{\Omega } w_1|\nabla u|^p \,dx\Bigr ) {\rm div} (w_1|\nabla u|^{p-2}\nabla u) = w_2f(u) &\text {in }\Omega , \\ u = 0 &\text {on }\partial \Omega , \end {cases} \] where $p\ge 2$, $\Omega $ is a $C^1$ domain of $\mathbb {R}^N$, $w_1, w_2$ are nonnegative functions, $m$ is a positive function and $f$ is an increasing one. Under some assumptions on $\Omega $, $w_1$, $w_2$, $m$ and $f$, we prove that the problem has no nontrivial stable solution in dimension $N \lt N^\#$. Moreover, additional assumptions on $\Omega $, $m$ or the boundedness of solutions can boost this critical dimension $N^\#$ to infinity.
