Multiplicity of solutions for a critical nonlinear Schrödinger–Kirchhoff-type equation

Abstract

In this paper, we study the following critical nonlinear Schrödinger–Kirchhoff equation: ($P$) $\begin{array}{r}\{\begin{array}{l}{\displaystyle -(a+b{\int }_{R^N}|\mathrm{\nabla u}{|}^2\mathrm{d}x)\mathrm{\Delta u}+V(x)u=P(x)|u{|}^{2^{\ast }-2}u+\mu |u{|}^{q-2}u,\mathrm{in}{\mathbb{R}}^N,}\\ u\in H^1({\mathbb{R}}^N)\end{array}\end{array}$($P$) where $a,b,\mu >0$, $N\ge 3$, $max\{2^{\ast }-1,2\}<q<2^{\ast }=\frac{2N}{N-2}$, $V(x)>0$ and $P(x)\ge 0$ are two continuous functions. By using the variational method and truncation technique, we prove the multiplicity of solutions for Equation (P).

Keywords:

• Schrödinger–Kirchhoff equation
• critical exponent
• local Pohozaev identities
• multiplicity of solutions

2020 Mathematics Subject Classifications:

• 35J10
• 47J30

Disclosure statement

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

Additional information

Funding

This work is supported by the National Natural Science Foundation of China [grant numbers 12261031, 12261076, 11801545] and the Fundamental Research Funds for the Central Universities [grant number 2023MS078].