#### Xian Hu, Yong-Yi Lan, Multiple solutions of Kirchhoff equations with a small perturbations, Vol. 2022 (2022), Article ID 19, pp. 1-11

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DOI: 10.23952/jnfa.2022.19

Received March 21, 2022; Accepted April 20, 2022; Published May 14, 2022

Abstract In this paper, we consider the following Kirchhoff equation

$-\bigg(a+b\int_{\Omega}|\nabla u|^{2}\,\mbox{d}x\bigg)\Delta u = f(x,u)+ tg(x,u), x\in \Omega,$
$u=0, x\in \partial \Omega,$

with the Dirichlet boundary value. Assuming that the main term $f(x,u)$ is sublinear and odd with respect to $u$, and the perturbation term is a any continuous function with a small coefficient, we establish the existence and multiplicity of nontrivial solutions for the problem. The approach relies on a combination of variational and minimization methods coupled with the reduction technique.