Kirchhoff–Boussinesq-type problems with positive and zero mass

Abstract

Using variational methods we show the existence of solutions for the following class of elliptic Kirchhoff–Boussinesq-type problems given by ${\mathrm{\Delta }}^{2}u-{\mathrm{\Delta }}_{p}u+u=h(u),\text{in}{\mathbb{R}}^N$ and ${\mathrm{\Delta }}^{2}u-{\mathrm{\Delta }}_{p}u=f(u),\text{in}{\mathbb{R}}^N,$ where $2<p\le \frac{2N}{N-2}$ for $N\ge 3$ and $2_{\ast \ast }=\mathrm{\infty }$ for N = 3, N = 4, $2_{\ast \ast }=\frac{2N}{N-4}$ for $N\ge 5$ and h and f are continuous functions that satisfy hypotheses considered by Berestycki and Lions [Nonlinear scalar field. Arch Rational Mech Anal. 1983;82:313–345]. More precisely, the problem with the nonlinearity h is related to the Positive mass case and the problem with the nonlinearity f is related to the Zero mass case. The main argument is to find a Palais–Smale sequence satisfying a property related to Pohozaev identity, as in Hirata et al. [Nonlinear scalar field equations in RN: mountain pass and symmetric mountain pass approaches. Topol Methods Nonlinear Anal. 2010;35:253–276], which was used for the first time by Jeanjean [On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer- type problem set on ${\mathbb{R}}^N$. Proc R Soc Edinb Sect A. 1999;129:787–809].

Keywords:

• Kirchhoff–Boussinesq
• zero mass
• positive mass

2010 Mathematics Subject Classifications:

• 35A15
• 35J20
• 35J61
• 58E05

Disclosure statement

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

Additional information

Funding

This work was supported by Federal District Research Support Foundation and Conselho Nacional de Desenvolvimento Científico e Tecnológico [grant number 316386/2021-9].