Existence of multiple solutions to Schrödinger–Poisson system in a nonlocal set up in $$\mathbb {R}^3$$

2021 ◽  
Vol 73 (1) ◽  
Author(s):  
Debajyoti Choudhuri ◽  
Kamel Saoudi
Keyword(s):  
Multiple Solutions ◽  
Poisson System ◽  
Set Up

scholarly journals Multiple solutions for a quasilinear Schrödinger–Poisson system

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Jing Zhang
Keyword(s):  
Nontrivial Solution ◽  
Ground State ◽  
Multiple Solutions ◽  
Continuous Functions ◽  
Ground State Solution ◽  
Nehari Manifold ◽  
State Solution ◽  
Poisson System ◽  
Subcritical Growth ◽  
Monotonicity Properties

AbstractIn this article, we consider the following quasilinear Schrödinger–Poisson system $$ \textstyle\begin{cases} -\Delta u+V(x)u-u\Delta (u^{2})+K(x)\phi (x)u=g(x,u), \quad x\in \mathbb{R}^{3}, \\ -\Delta \phi =K(x)u^{2}, \quad x\in \mathbb{R}^{3}, \end{cases} $$ { − Δ u + V ( x ) u − u Δ ( u 2 ) + K ( x ) ϕ ( x ) u = g ( x , u ) , x ∈ R 3 , − Δ ϕ = K ( x ) u 2 , x ∈ R 3 , where $V,K:\mathbb{R}^{3}\rightarrow \mathbb{R}$ V , K : R 3 → R and $g:\mathbb{R}^{3}\times \mathbb{R}\rightarrow \mathbb{R}$ g : R 3 × R → R are continuous functions; g is of subcritical growth and has some monotonicity properties. The purpose of this paper is to find the ground state solution of (0.1), i.e., a nontrivial solution with the least possible energy by taking advantage of the generalized Nehari manifold approach, which was proposed by Szulkin and Weth. Furthermore, infinitely many geometrically distinct solutions are gained while g is odd in u.

scholarly journals Multiple Solutions for a Class of Fractional Schrödinger-Poisson System

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Lizhen Chen ◽  
Anran Li ◽  
Chongqing Wei
Keyword(s):  
Variational Methods ◽  
Multiple Solutions ◽  
Mountain Pass Theorem ◽  
Negative Energy ◽  
Mountain Pass ◽  
Poisson System ◽  
Fountain Theorem ◽  
Symmetric Mountain Pass Theorem ◽  
Dual Fountain Theorem

We investigate a class of fractional Schrödinger-Poisson system via variational methods. By using symmetric mountain pass theorem, we prove the existence of multiple solutions. Moreover, by using dual fountain theorem, we prove the above system has a sequence of negative energy solutions, and the corresponding energy values tend to 0. These results extend some known results in previous papers.

scholarly journals New multiple solutions for a Schrödinger–Poisson system involving concave-convex nonlinearities

2020 ◽  
Vol 44 (3) ◽  
pp. 986-997
Author(s):  
Chun-Yu LEI ◽  
Gao-Sheng LIU ◽  
Chang-Mu CHU ◽  
Hong-Min SUO
Keyword(s):  
Multiple Solutions ◽  
Poisson System
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