scholarly journals Existence of Solutions for a Schrödinger–Poisson System with Critical Nonlocal Term and General Nonlinearity

2020 ◽  
Vol 2020 ◽  
pp. 1-5
Author(s):  
Jiafeng Zhang ◽  
Wei Guo ◽  
Changmu Chu ◽  
Hongmin Suo
Keyword(s):  
Variational Method ◽  
Existence Of Solutions ◽  
Nonlocal Term ◽  
Poisson System ◽  
Nontrivial Solutions ◽  
Existence And Multiplicity ◽  
Analysis Technique ◽  
General Nonlinearity

We study the existence and multiplicity of nontrivial solutions for a Schrödinger–Poisson system involving critical nonlocal term and general nonlinearity. Based on the variational method and analysis technique, we obtain the existence of two nontrivial solutions for this system.

scholarly journals Existence and multiplicity of solutions for Kirchhof-type problems with Sobolev–Hardy critical exponent

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Hongsen Fan ◽  
Zhiying Deng
Keyword(s):  
Variational Method ◽  
Boundary Value Problem ◽  
Critical Exponent ◽  
Positive Solution ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Elliptic Boundary Value Problem ◽  
Nontrivial Solutions ◽  
Elliptic Boundary Value ◽  
Existence And Multiplicity

AbstractIn this paper, we discuss a class of Kirchhof-type elliptic boundary value problem with Sobolev–Hardy critical exponent and apply the variational method to obtain one positive solution and two nontrivial solutions to the problem under certain conditions.

scholarly journals Existence and Multiplicity of Nontrivial Solutions for a Class of Semilinear Fractional Schrödinger Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-7 ◽  
Author(s):  
Xinsheng Du ◽  
Anmin Mao
Keyword(s):  
Existence Of Solutions ◽  
High Energy ◽  
Sufficient Condition ◽  
Schrödinger Equations ◽  
Nontrivial Solutions ◽  
Existence And Multiplicity ◽  
Fractional Schrödinger Equations ◽  
Fountain Theorems ◽  
Superquadratic Growth ◽  
Variant Fountain Theorems

This paper is concerned with the existence of solutions to the following fractional Schrödinger type equations: -∆su+Vxu=fx,u,  x∈RN, where the primitive of the nonlinearity f is of superquadratic growth near infinity in u and the potential V is allowed to be sign-changing. By using variant Fountain theorems, a sufficient condition is obtained for the existence of infinitely many nontrivial high energy solutions.

scholarly journals Existence of Solutions for a Robin Problem Involving thep(x)-Laplace Operator

2016 ◽  
Vol 2016 ◽  
pp. 1-8 ◽  
Author(s):  
Mostafa Allaoui
Keyword(s):  
Variational Method ◽  
Boundary Value Problem ◽  
Laplace Operator ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Robin Problem ◽  
Multiplicity Of Solutions ◽  
Existence And Multiplicity ◽  
Robin Boundary ◽  
Robin Boundary Value Problem

In this article we study the nonlinear Robin boundary-value problem-Δp(x)u=f(x,u)  in  Ω,|∇u|px-2(∂u/∂ν)+β(x)up(x)-2u=0on∂Ω. Using the variational method, under appropriate assumptions onf, we obtain results on existence and multiplicity of solutions.

scholarly journals On the Existence of Solutions for the Critical Fractional Laplacian Equation inℝN

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Zifei Shen ◽  
Fashun Gao
Keyword(s):  
Variational Method ◽  
Critical Exponent ◽  
Lower Order ◽  
Fractional Laplacian ◽  
Critical Power ◽  
Existence Of Solutions ◽  
Order Perturbation ◽  
Potential Well ◽  
Nontrivial Solutions ◽  
Laplacian Equation

We study existence of solutions for the fractional Laplacian equation-Δsu+Vxu=u2*s-2u+fx, uinℝN,u∈Hs(RN), with critical exponent2*s=2N/(N-2s),N>2s,s∈0, 1, whereVx≥0has a potential well andf:ℝN×ℝ→ℝis a lower order perturbation of the critical poweru2*s-2u. By employing the variational method, we prove the existence of nontrivial solutions for the equation.

scholarly journals Multiple positive solutions for a logarithmic Schrödinger–Poisson system with singular nonelinearity

2021 ◽  
pp. 1-15
Author(s):  
Linyan Peng ◽  
Hongmin Suo ◽  
Deke Wu ◽  
Hongxi Feng ◽  
Chunyu Lei
Keyword(s):  
Variational Method ◽  
Critical Point ◽  
Positive Solutions ◽  
Critical Point Theory ◽  
Point Theory ◽  
Multiple Positive Solutions ◽  
Poisson System ◽  
Smooth Bounded Domain ◽  
Existence And Multiplicity ◽  
Multiplicity Of Positive Solutions

In this article, we devote ourselves to investigate the following logarithmic Schrödinger–Poisson systems with singular nonlinearity { − Δ u + ϕ u = | u | p−2 u log ⁡ | u | + λ u γ , i n   Ω , − Δ ϕ = u 2 , i n   Ω , u = ϕ = 0 , o n   ∂ Ω , where Ω is a smooth bounded domain with boundary 0 < γ < 1 , p ∈ ( 4 , 6 ) and λ > 0 is a real parameter. By using the critical point theory for nonsmooth functional and variational method, the existence and multiplicity of positive solutions are established.

scholarly journals Existence of Multiple Nontrivial Solutions for a Strongly Indefinite Schrödinger-Poisson System

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Shaowei Chen ◽  
Liqin Xiao
Keyword(s):  
Nontrivial Solution ◽  
Infinitely Many Solutions ◽  
Poisson System ◽  
Nontrivial Solutions ◽  
Local Linking ◽  
Variational Functional ◽  
Indefinite Potential ◽  
Fountain Theorems ◽  
General Nonlinearity ◽  
Odd Nonlinearity

We consider a Schrödinger-Poisson system inℝ3with a strongly indefinite potential and a general nonlinearity. Its variational functional does not satisfy the global linking geometry. We obtain a nontrivial solution and, in case of odd nonlinearity, infinitely many solutions using the local linking and improved fountain theorems, respectively.

Existence and multiplicity of solutions for Kirchhoff–Schrödinger–Poisson system with critical growth

2021 ◽  
Author(s):  
Guofeng Che ◽  
Haibo Chen
Keyword(s):  
Variational Methods ◽  
Critical Growth ◽  
Poisson System ◽  
Nontrivial Solutions ◽  
Multiplicity Of Solutions ◽  
Existence And Multiplicity

This paper is concerned with the following Kirchhoff–Schrödinger–Poisson system: [Formula: see text] where constants [Formula: see text], [Formula: see text] and [Formula: see text] are the parameters. Under some appropriate assumptions on [Formula: see text], [Formula: see text] and [Formula: see text], we prove the existence and multiplicity of nontrivial solutions for the above system via variational methods. Some recent results from the literature are greatly improved and extended.

scholarly journals Existence of solutions for an elliptic p(x)-Kirchhoff-type systems in unbounded domain

2018 ◽  
Vol 36 (3) ◽  
pp. 193-205 ◽  
Author(s):  
Abdelamlek Brahim ◽  
Ali Djellit ◽  
Saadia Tas
Keyword(s):  
Variational Method ◽  
Elliptic System ◽  
Unbounded Domain ◽  
Existence Of Solutions ◽  
Mountain Pass Theorem ◽  
Main Tool ◽  
Type Systems ◽  
Mountain Pass ◽  
Nonlocal Term ◽  
Kirchhoff Type

In this paper we study of the existence of solutions for a class of elliptic system with nonlocal term in R^{N}. The main tool used is the variational method, more precisely, the Mountain Pass Theorem.

scholarly journals Existence of Solutions for Choquard Type Elliptic Problems with Doubly Critical Nonlinearities

2019 ◽  
Vol 21 (1) ◽  
pp. 77-93
Author(s):  
Yansheng Shen
Keyword(s):  
Variational Methods ◽  
Minimization Problem ◽  
Existence Of Solutions ◽  
Elliptic Problems ◽  
Type Equation ◽  
Nontrivial Solutions ◽  
Critical Nonlinearities

Abstract In this article, we first study the existence of nontrivial solutions to the nonlocal elliptic problems in ℝ N {\mathbb{R}^{N}} involving fractional Laplacians and the Hardy–Sobolev–Maz’ya potential. Using variational methods, we investigate the attainability of the corresponding minimization problem, and then obtain the existence of solutions. We also consider another Choquard type equation involving the p-Laplacian and critical nonlinearities in ℝ N {\mathbb{R}^{N}} .

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