This paper is concerned with the existence of positive solutions to Schrödinger-Kirchhoff-type equations $\left(\mathcal{P}\right)\left\{\begin{array}{ll}-\left(a+b{\int }_{{\mathbb{R}}^3}|\mathrm{\nabla }u{|}^2\right)\mathrm{\Delta }u+V\left(x\right)u=|u{|}^{p-1}u& \mathrm{i}\mathrm{n}{\mathbb{R}}^3,\\ u\in H^1\left({\mathbb{R}}^3\right),\end{array}\right.$ where a and b are two positive constants, $p\in (1,5)$ and $V:{\mathbb{R}}^3\to \mathbb{R}$ is a potential function. Under certain assumptions on V, we prove that $\left(\mathcal{P}\right)$ has no ground state solution. However, we can show $\left(\mathcal{P}\right)$ has a positive bound state solution by applying a new version of the global compactness lemma and the linking theorem.

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具有反势的薛定谔-基尔霍夫方程的正解

本文关注薛定谔-基尔霍夫型方程正解的存在性$\left(\mathcal{磷}\right)\left\{\begin{array}{ll}-\left(\mathrm{一种}+b{\int }_{{\mathbb{R}}^3}|\mathrm{\nabla }你{|}^2\right)\mathrm{\Delta }你+五\left(X\right)你=|你{|}^{p-1}你& \mathrm{一世}\mathrm{n}{\mathbb{R}}^3,\\ 你\in H^1\left({\mathbb{R}}^3\right),\end{array}\right.$其中a和b是两个正常数，$p\in (1,5)$和$五：{\mathbb{R}}^3\to \mathbb{R}$是一个势函数。在对V的某些假设下，我们证明$\left(\mathcal{磷}\right)$没有基态解。但是，我们可以展示$\left(\mathcal{磷}\right)$通过应用新版本的全局紧致引理和链接定理，得到正的束缚态解。