Kirchhoff-Boussinesq-type problems with positive and zero mass

Using variational methods we show the existence of solutions for the following class of elliptic Kirchhoff-Boussinesq-type problems given by delta(2)u - delta pu + u = h(u), in R(N)and delta(2)u - delta(p)u = f(u), in R-N,where 2 < p <= 2N/N-2 for N >= 3 and 2(& lowast;& lowast;) = infinity for N = 3, N = 4, 2(& lowast;& lowast; )= 2N/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 non linearity 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 RN. Proc R Soc Edinb Sect A. 1999;129:787-809].

Automated summary

Using variational methods, this study proves the existence of solutions for two classes of elliptic Kirchhoff-Boussinesq-type problems in R(N) and R-N, given by delta(2)u - delta pu + u = h(u) and delta(2)u - delta(p)u = f(u), respectively.