scholarly journals Multiple positive bound states for critical Schrödinger-Poisson systems

2019 ◽  
Vol 25 ◽  
pp. 73 ◽  
Author(s):  
Giovanna Cerami ◽  
Riccardo Molle
Keyword(s):  
Variational Methods ◽  
Ground State ◽  
Bound States ◽  
Nonlinear Term ◽  
Critical Growth ◽  
Poisson System ◽  
Lack Of Compactness ◽  
Ground State Solutions ◽  
Existence And Multiplicity

Using variational methods we prove some results about existence and multiplicity of positive bound states of to the following Schrödinger-Poisson system: [see formula in PDF] We remark that (SP) exhibits a “double” lack of compactness because of the unboundedness of ℝ3 and the critical growth of the nonlinear term and that in our assumptions ground state solutions of (SP) do not exist.

scholarly journals Ground state solutions for Kirchhoff-type equations with a general nonlinearity in the critical growth

2018 ◽  
Vol 7 (4) ◽  
pp. 535-546 ◽  
Author(s):  
Li-Ping Xu ◽  
Haibo Chen
Keyword(s):  
Variational Methods ◽  
Ground State ◽  
Critical Growth ◽  
Ground State Solutions ◽  
Kirchhoff Type ◽  
General Nonlinearity

AbstractIn this paper, we concern ourselves with the following Kirchhoff-type equations:\left\{\begin{aligned} \displaystyle-\biggl{(}a+b\int_{\mathbb{R}^{3}}\lvert% \nabla u\rvert^{2}\,dx\biggr{)}\triangle u+Vu&\displaystyle=f(u)\quad\text{in % }\mathbb{R}^{3},\\ \displaystyle u&\displaystyle\in H^{1}(\mathbb{R}^{3}),\end{aligned}\right.where a, b and V are positive constants and f has critical growth. We use variational methods to prove the existence of ground state solutions. In particular, we do not use the classical Ambrosetti–Rabinowitz condition. Some recent results are extended.

scholarly journals Concentration results for a magnetic Schrödinger-Poisson system with critical growth

2020 ◽  
Vol 10 (1) ◽  
pp. 775-798
Author(s):  
Jingjing Liu ◽  
Chao Ji
Keyword(s):  
Variational Methods ◽  
Nonlinear Term ◽  
Type Equation ◽  
Critical Growth ◽  
Poisson System ◽  
Nontrivial Solutions ◽  
Penalization Technique ◽  
Poisson Type

Abstract This paper is concerned with the following nonlinear magnetic Schrödinger-Poisson type equation $$\begin{array}{} \displaystyle \left\{ \begin{aligned} &\Big(\frac{\epsilon}{i}\nabla-A(x)\Big)^{2}u+V(x)u+\epsilon^{-2}(\vert x\vert^{-1}\ast \vert u\vert^{2})u=f(|u|^{2})u+\vert u\vert^{4}u \quad \hbox{in }\mathbb{R}^3,\\ &u\in H^{1}(\mathbb{R}^{3}, \mathbb{C}), \end{aligned} \right. \end{array}$$ where ϵ > 0, V : ℝ3 → ℝ and A : ℝ3 → ℝ3 are continuous potentials, f : ℝ → ℝ is a subcritical nonlinear term and is only continuous. Under a local assumption on the potential V, we use variational methods, penalization technique and Ljusternick-Schnirelmann theory to prove multiplicity and concentration of nontrivial solutions for ϵ > 0 small.

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.

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