A remark on Kirchhoff-type equations in ℝ⁴ involving critical growth

ABSTRACT

We consider the following Kirchhoff type equation: $-\left(a+b{\int }_{\mathrm{\Omega }}|\mathrm{\nabla }u{|}^2\mathrm{d}x\right)\mathrm{\Delta }u=g\left(u\right),\mathrm{i}\mathrm{n}{\mathbb{R}}^4,$ where a, b are positive constants and $g\in C(\mathbb{R},\mathbb{R})$. Under the critical growth assumptions on g, we obtain the existence of least energy solutions by studying the associated minimization problem with a new constraint. The mountain pass characterization of least energy solutions is also investigated without the monotonicity assumption on $g\left(s\right)/s$. Our results present some new observations.

Keywords:

• Kirchhof- type equation
• critical growth
• least energy solution
• Pohozaev identity

AMS Subject Classifications:

• Primary: 35J20
• 35B33
• Secondary: 35B09
• 35J60

Disclosure statement

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

Additional information

Funding

Supported by Natural Science Foundation of China [grant numbers 11901418, 11771319 and 11971339].