Infinitely many solutions for a class of superlinear Dirac–Poisson system
Abstract This paper is concerned with the nonlinear Dirac–Poisson system − i ∑ k = 1 3 α k ∂ k u + ( V ( x ) + a ) β u + ω u − ϕ u = F u ( x , u ) , − Δ ϕ = 4 π | u | 2 , in R 3 where V is an external potential and F is a superlinear nonlinearity modeling various types of interactions. Existence and multiplicity of stationary solutions are obtained for the system with periodicity condition via variational methods.
The main objective of this article is to study dynamic of the ::: three-dimensional Boussinesq equations with the periodic boundary ::: condition.We prove that when the Rayleigh number $R$ crosses the ::: first critical Rayleigh number $R_c$, the Rayleigh-Benard problem ::: bifurcates from the basic state to an global attractor $\Sigma$, which is homeomorphic to $S^3$.