This paper mainly considers a kind of nonlinear elliptic equation with a Kirchhoff type nonlocal term
$\begin{equation} -\left(a+b\int_{\mathbb{R}^3}\left| \nabla u\right|^2 \right)\Delta u+V(x)u=Q(x)\left| u\right|^{p-1}u, \quad x\in\mathbb{R}^3, \end{equation}$(0.1)
where$ a,b>0 $are constants,$ p\in(1,5) $,$ V(x) $and$ Q(x) $are$ L^\infty(\mathbb{R}^3) $functions. It is known that if we apply the mountain pass lemma directly to obtain solution (i.e., mountain pass solution) to the equation (0.1), we must require$ 3\le p<5 $because of the appearance of nonlocal terms. When$ p\in(1,3) $, the fundamental difficulty in applying the mountain pass lemma is that we are unable to verify the boundedness of the (PS) sequence. To overcome this difficulty, when$ Q(x)\equiv 1 $, paper [Acta Math Sci, 2025, 45B(2): 385-400] introduced a new technique to demonstrate the equation (0.1) exists a mountain pass solution for all$ p\in(1,5) $, and discussed the relationship between the mountain pass solution and the ground state solution obtained. The purpose of this paper is to extend the results of [Acta Math Sci, 2025, 45B(2): 385-400] to the general case$ Q(x)\not\equiv 1 $.
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