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

Abstract

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), \quad \mbox{in} \mathbb{R}^3, \end{equation*} $ where $ a > 0, b\geq 0 $ are two constants and $ V:{\mathbb{R}}^{3}\rightarrow {\mathbb{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.