ABSTRACT

In this paper, we focus on the existence of multiplicity of solutions for the singular quasilnear Schrödinger–Kirchhoff type problem: $\begin{array}{rl}& -\left(a+b{\int }_{{\mathbb{R}}^3}\left(1+\frac{{\alpha }^2}{2}|u{|}^{2(\alpha -1)}\right)|\mathrm{\nabla }u{|}^2\mathrm{d}x\right)\left(\mathrm{\Delta }u+\frac{\alpha }{2}\mathrm{\Delta }\left(|u{|}^{\alpha }\right)|u{|}^{\alpha -2}u\right)\\ & =\lambda V\left(x\right)|u{|}^{p-2}u+G\left(x\right)|u{|}^{4}u,\end{array}$where $x\in {\mathbb{R}}^3$, a>0, $b\ge 0$, $0<\alpha <1$, 1<p<2, $\lambda \in \mathbb{R}$, $0\le V\left(x\right)\in C\left({\mathbb{R}}^3\right)\cap L^q\left({\mathbb{R}}^3\right)$ with $q=\frac{6}{6-p}$, $G\left(x\right)\in C\left({\mathbb{R}}^3\right)\cap L^{\mathrm{\infty }}\left({\mathbb{R}}^3\right)$, and we get the existence of infinitely many weak solutions by variational methods.

中文翻译：

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一些具有临界指数的奇异拟线性薛定谔-基尔霍夫方程的解的多重性

摘要

在本文中，我们关注奇异拟似薛定谔-基尔霍夫型问题的多重解的存在性：$\begin{array}{rl}& -\left(\mathrm{一个}+b{\int }_{{\mathbb{R}}^3}\left(1+\frac{{\alpha }^2}{2}|你{|}^{2(\alpha -1)}\right)|\mathrm{\nabla }你{|}^2\mathrm{d}X\right)\left(\mathrm{\Delta }你+\frac{\alpha }{2}\mathrm{\Delta }\left(|你{|}^{\alpha }\right)|你{|}^{\alpha -2}你\right)\\ & =\lambda 五\left(X\right)|你{|}^{p-2}你+G\left(X\right)|你{|}^4你,\end{array}$在哪里$X\in {\mathbb{R}}^3$, a >0,$b\ge 0$,$0<\alpha <1$, 1< p <2,$\lambda \in \mathbb{R}$,$0\le 五\left(X\right)\in C\left({\mathbb{R}}^3\right)\cap {\mathrm{大号}}^q\left({\mathbb{R}}^3\right)$和$q=\frac{6}{6-p}$,$G\left(X\right)\in C\left({\mathbb{R}}^3\right)\cap {\mathrm{大号}}^{\mathrm{\infty }}\left({\mathbb{R}}^3\right)$，我们通过变分方法得到无限多个弱解的存在性。