Ground state solutions for asymptotically periodic Schrödinger–Poisson systems involving Hartree-type nonlinearities

Existence of Positive Ground State Solutions for a Class of Asymptotically Periodic Schrödinger-poisson Systems

In this article, by using variational method, we study the existence of a positive ground state solution for the Schrödinger-Poisson system −∆u+ V (x)u+K(x)φu = f(x, u), x ∈ R, −∆φ = K(x)u, x ∈ R, where V (x),K(x) and f(x, u) are asymptotically periodic functions in x at infinity.

Positive solutions for asymptotically periodic Kirchhoff-type equations with critical growth

In this paper‎, ‎we consider the following Kirchhoff-type equations‎: ‎$-‎left(a+bint_{mathbb{R}^{3}}|nabla u|^{2}right)Delta u+V(x) u=lambda$ $f(x,u)+u^{5}‎, ‎quad mbox{in }mathbb{R}^{3},$ ‎$u(x)>0‎, ‎quad mbox{in }mathbb{R}^{3},$ ‎$uin H^{1}(mathbb{R}^{3})‎ ,‎$ ‎ ‎‎‎where $a,b>0$ are constants and $lambda$ is a positive parameter‎. ‎The aim of this paper is to study the existence of positive ...

Ground State Solutions for the Periodic Discrete Nonlinear Schrödinger Equations with Superlinear Nonlinearities

and Applied Analysis 3 In this paper, we also consider themultiplicity of solutions of (7). For each l ∈ Z, let l ∗ u = (u n+lT ) n∈Z , ∀u = (u n ) n∈Z , (21) which defines a Z-action on E. By the periodicity of the coefficients, we know that both J and J󸀠 are Z-invariants. Therefore, ifu ∈ E is a critical point of J, so is l∗u. Two critical points u 1 , u 2 ∈ E of J are said to be geometricall...

Ground state solutions for an asymptotically periodic and superlinear Schrödinger equation

We consider the semilinear Schrödinger equation { −4 u+ V (x)u = f(x, u), x ∈ RN , u ∈ H1(RN ), where V (x) is asymptotically periodic and sign-changing, f(x, u) is a superlinear, subcritical nonlinearity. Under asymptotically periodic V (x) and a super-quadratic condition about f(x, u). We prove that the above problem has a ground state solution which minimizes the corresponding energy among a...

Existence of Periodic Solutions for a Class of Asymptotically p-Linear Discrete Systems Involving p-Laplacian