Multiplicity of solutions for some singular quasilinear Schrödinger–Kirchhoff equations with critical exponents

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.

Keywords:

• Schrödinger–Kirchhoff type equation
• critical exponent
• concentration-compactness

AMS Subject Classifications:

• 35A15
• 35B33
• 35J60

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work was supported by the National Natural Science Foundation of China [grant number 11171220].