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.

Justification of the Lugiato-Lefever Model from a Damped Driven ϕ4 Equation

The Lugiato-Lefever equation is a damped and driven version of the well-known nonlinear Schrödinger equation. It is a mathematical model describing complex phenomena in dissipative and nonlinear optical cavities. Within the last two decades, the equation has gained much attention as it has become the basic model describing microresonator (Kerr) frequency combs. Recent works derive the Lugiato-Lefever equation from a class of damped driven ϕ 4 equations closed to resonance. In this paper, we provide a justification of the envelope approximation. From the analysis point of view, the result is novel and non-trivial as the drive yields a perturbation term that is not square integrable. The main approach proposed in this work is to decompose the solutions into a combination of the background and the integrable component. This paper is the first part of a two-manuscript series.

Existence of Solutions for Nonhomogeneous Choquard Equations Involving p-Laplacian

This paper is devoted to investigating a class of nonhomogeneous Choquard equations with perturbation involving p-Laplacian. Under suitable hypotheses about the perturbation term, the existence of at least two nontrivial solutions for the given problems is obtained using Nehari manifold and minimax methods.

On the noncommutative energy level in a two-dimensional anharmonic oscillatoric Oscillator

We study quantum properties of a two-dimensional anharmonic oscillator in the space-space and momentum-momentum in noncommutative variables. This work show ex- plicitly the effects of both deformations in the energy levels. The perturbation term in the Hamiltonian manifest the main difference of the noncommutative parameters. Particular nu- merical values of noncommutative parameters are examined and graphically illustrated for different nx and ny non-negative integers.

On the dynamics of a class of perturbed cyclic Lotka-Volterra systems

We consider a special class of Lotka–Volterra systems where the associated interaction matrix is cyclic, but asymmetric, with a perturbation term on each row. After some discussion of the dynamics under a general setting, we focus our attention on 3D systems for a more detailed study. We derive sufficient conditions for the existence and stability of the nontrivial interior equilibrium. We also show that Hopf bifurcation occurs when the size of the perturbation is large. Such analysis can be similarly extended to higher dimensional systems, and we mention some results in 4D case.