Existence of solutions for modified Kirchhoff-type equation without the Ambrosetti-Rabinowitz condition

• Main Authors: Zhongxiang Wang, Gao Jia
• Format: Article
• Language: English
• Published: AIMS Press 2021-02-01
• Series: AIMS Mathematics
• Subjects: modified kirchhoff-type equation; ground state solution; nehari manifold; pohozaev identity
• Online Access: http://www.aimspress.com/article/doi/10.3934/math.2021272?viewType=HTML

This paper is devoted to studying a class of modified Kirchhoff-type equations \begin{equation*} -\Big(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}dx\Big)\Delta u+V(x)u-u\Delta(u^2)=f(x,u),\hspace{0.5cm} \mbox{in}\ \mathbb{R}^3, \end{equation*} where $a>0, b\geq 0$ are two constants and $V:\R^{3}\rightarrow \R$ is a potential function. The existence of non-trivial solution to the above problem is obtained by the perturbation methods. Moreover, when $u>0$ and $f(x,u)=f(u)$, under suitable hypotheses on $V(x)$ and $f(u)$, we obtain the existence of a positive ground state solution by using a monotonicity trick and a new version of global compactness lemma. The character of this work is that for $f(u)\sim|u|^{p-2}u$ we prove the existence of a positive ground state solution in the case where $p\in(2,3]$, which has few results for the modified Kirchhoff equation. Hence our results improve and extend the existence results in the related literatures.