Infinitely Many Solutions for the Nonlinear Schrödinger–Poisson System with Broken Symmetry

In this paper, we consider the following Schrödinger–Poisson system with perturbation: { - Δ ⁢ u + u + λ ⁢ ϕ ⁢ ( x ) ⁢ u = | u | p - 2 ⁢ u + g ⁢ ( x ) , x ∈ ℝ 3 , - Δ ⁢ ϕ = u 2 , x ∈ ℝ 3 , \left\{\begin{aligned} \displaystyle-\Delta u+u+\lambda\phi(x)u&\displaystyle=% |u|^{p-2}u+g(x),&&\displaystyle x\in\mathbb{R}^{3},\\ \displaystyle-\Delta\phi&\displaystyle=u^{2},&&\displaystyle x\in\mathbb{R}^{3% },\end{aligned}\right. where λ > 0 {\lambda>0} , p ∈ ( 3 , 6 ) {p\in(3,6)} and the radial general perturbation term g ⁢ ( x ) ∈ L p p - 1 ⁢ ( ℝ 3 ) {g(x)\in L^{\frac{p}{p-1}}(\mathbb{R}^{3})} . By establishing a new abstract perturbation theorem based on the Bolle’s method, we prove the existence of infinitely many radial solutions of the above system. Moreover, we give the asymptotic behaviors of these solutions as λ → 0 {\lambda\to 0} . Our results partially solve the open problem addressed in [Y. Jiang, Z. Wang and H.-S. Zhou, Multiple solutions for a nonhomogeneous Schrödinger–Maxwell system in ℝ 3 \mathbb{R}^{3} , Nonlinear Anal. 83 2013, 50–57] on the existence of infinitely many solutions of the Schrödinger–Poisson system for p ∈ ( 2 , 4 ] {p\in(2,4]} and a general perturbation term g.

Stability of Contact Discontinuity with General Perturbation for the Compressible Navier-Stokes Equations with Reaction Diffusion

In this article,for the compressible Navier-Stokes equations which have reaction diffusion, the stability of contact discontinuities is considered. The new characteristic for the flow is appearance of the divergence between energy gained and lost because of the reaction . In the energy equations,the term related to the mass fraction of the reactant leads to new technical problem. To solve this problem, in terms of the solutions,a new system should be set up. Using the anti-derivative method and the elaborated energy method, we obtain that as long as the general perturbation of the initial datum plane and the strength of the contact wave are properly small, the contact wave is nonlinear and stable. As a byproduct, we can establish the convergence velocity of contact wave.

Periodic orbits in the restricted problem of three bodies in a three-dimensional coordinate system when the smaller primary is a triaxial rigid body

In this paper, we have studied the equations of motion for the problem, which are regularised in the neighbourhood of one of the finite masses and the existence of periodic orbits in a three-dimensional coordinate system when μ = 0. Finally, it establishes the canonical set (l, L, g, G, h, H) and forms the basic general perturbation theory for the problem.