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 Δ 2 u − Δ p u + u = h ( u ) , in R N and Δ 2 u − Δ p u = f ( u ) , in R N , where 2 < p ≤ 2 N N − 2 for N ≥ 3 and 2 ∗ ∗ = ∞ for N = 3, N = 4, 2 ∗ ∗ = 2 N N − 4 for N ≥ 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 R N . Proc R Soc Edinb Sect A. 1999;129:787–809].