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Instability of solutions to Kirchhoff type problems in low dimension

Volume 124 / 2020

Nhat Vy Huynh, Phuong Le 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.

Authors

  • Nhat Vy HuynhFaculty of Mathematics and Computer Science
    University of Science
    Vietnam National University – Ho Chi Minh City
    Ho Chi Minh City, Vietnam
    and
    Department of Fundamental Sciences
    Ho Chi Minh City University of Transport
    Ho Chi Minh City, Vietnam
  • Phuong LeDivision of Computational Mathematics and Engineering
    Institute for Computational Science
    Ton Duc Thang University
    Ho Chi Minh City, Vietnam
    and
    Faculty of Mathematics and Statistics
    Ton Duc Thang University
    Ho Chi Minh City, Vietnam
    e-mail

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