scholarly journals Ground states for critical fractional Schr\”{o}dinger-Poisson systems with vanishing potentials

2021 ◽  
Author(s):  
Xilin Dou ◽  
xiaoming he
Keyword(s):  
Ground State ◽  
Mountain Pass Theorem ◽  
Ground State Solution ◽  
Nonlocal Term ◽  
Concentration Compactness ◽  
Poisson System ◽  
Concentration Compactness Principle ◽  
Whole Space ◽  
Compactness Principle ◽  
Vanishing Potentials

This paper deals with a class of fractional Schr\”{o}dinger-Poisson system \[\begin{cases}\displaystyle (-\Delta )^{s}u+V(x)u-K(x)\phi |u|^{2^*_s-3}u=a (x)f(u), &x \in \R^{3}\\ (-\Delta )^{s}\phi=K(x)|u|^{2^*_s-1}, &x \in \R^{3}\end{cases} \]with a critical nonlocal term and multiple competing potentials, which may decay and vanish at infinity, where $s \in (\frac{3}{4},1)$, $ 2^*_s = \frac{6}{3-2s}$ is the fractional critical exponent. The problem is set on the whole space and compactness issues have to be tackled. By employing the mountain pass theorem, concentration-compactness principle and approximation method, the existence of a positive ground state solution is obtained under appropriate assumptions imposed on $V, K, a$ and $f$.

scholarly journals Existence of positive ground state solutions of Schrödinger–Poisson system involving negative nonlocal term and critical exponent on bounded domain

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Wenxuan Zheng ◽  
Wenbin Gan ◽  
Shibo Liu
Keyword(s):  
Critical Exponent ◽  
Bounded Domain ◽  
Ground State ◽  
Mountain Pass Theorem ◽  
Nonlocal Term ◽  
Concentration Compactness ◽  
Poisson System ◽  
Ground State Solutions ◽  
Concentration Compactness Principle ◽  
Compactness Principle

AbstractIn this paper, we prove the existence of positive ground state solutions of the Schrödinger–Poisson system involving a negative nonlocal term and critical exponent on a bounded domain. The main tools are the mountain pass theorem and the concentration compactness principle.

scholarly journals Schrödinger equations with asymptotically periodic terms

2015 ◽  
Vol 145 (4) ◽  
pp. 745-757 ◽  
Author(s):  
Reinaldo de Marchi
Keyword(s):  
Mountain Pass Theorem ◽  
Ground State Solution ◽  
Schrodinger Equations ◽  
Periodic Case ◽  
Local Version ◽  
Concentration Compactness ◽  
Schrödinger Equations ◽  
Concentration Compactness Principle ◽  
Asymptotically Periodic ◽  
Compactness Principle

We study the existence of non-trivial solutions for a class of asymptotically periodic semilinear Schrödinger equations in ℝN. By combining variational methods and the concentration-compactness principle, we obtain a non-trivial solution for the asymptotically periodic case and a ground state solution for the periodic one. In the proofs we apply the mountain pass theorem and its local version.

scholarly journals A perturbed nonlinear elliptic PDE with two Hardy–Sobolev critical exponents

2016 ◽  
Vol 18 (04) ◽  
pp. 1550061 ◽  
Author(s):  
X. Zhong ◽  
W. Zou
Keyword(s):  
Critical Exponents ◽  
Mean Curvature ◽  
Ground State Solution ◽  
Nehari Manifold ◽  
Concentration Compactness ◽  
Nonlinear Elliptic ◽  
Concentration Compactness Principle ◽  
The Mean ◽  
Compactness Principle ◽  
The Nehari Manifold

Let [Formula: see text] be a [Formula: see text] open bounded domain in [Formula: see text] ([Formula: see text]) with [Formula: see text]. Suppose that [Formula: see text] is [Formula: see text] at [Formula: see text] and the mean curvature of [Formula: see text] at [Formula: see text] is negative. Consider the following perturbed PDE involving two Hardy–Sobolev critical exponents: [Formula: see text] where [Formula: see text]. The existence of ground state solution is studied under different assumptions via the concentration compactness principle and the Nehari manifold method. We also apply a perturbation method to study the existence of positive solution.

Multiplicity of solutions to the weighted critical quasilinear problems

2012 ◽  
Vol 55 (1) ◽  
pp. 181-195 ◽  
Author(s):  
Sihua Liang ◽  
Jihui Zhang
Keyword(s):  
Bounded Domain ◽  
Mountain Pass Theorem ◽  
Mountain Pass ◽  
Concentration Compactness ◽  
Positive Parameter ◽  
Symmetric Mountain Pass Theorem ◽  
Concentration Compactness Principle ◽  
Compactness Principle ◽  
Quasilinear Problems ◽  
Image Position

AbstractWe consider a class of critical quasilinear problemswhere 0 ∈ Ω ⊂ ℝN, N ≥ 3, is a bounded domain and 1 < p < N, a < N/p, a ≤ b < a + 1, λ is a positive parameter, 0 ≤ μ < $\bar{\mu}$ ≡ ((N − p)/p − a)p, q = q*(a, b) ≡ Np/[N − pd] and d ≡ a+1 − b. Infinitely many small solutions are obtained by using a version of the symmetric Mountain Pass Theorem and a variant of the concentration-compactness principle. We deal with a problem that extends some results involving singularities not only in the nonlinearities but also in the operator.

Concentration-Compactness Principle of Singular Trudinger--Moser Inequalities in ℝ n and n-Laplace Equations

2018 ◽  
Vol 18 (3) ◽  
pp. 567-585 ◽  
Author(s):  
Caifeng Zhang ◽  
Lu Chen
Keyword(s):  
Positive Constant ◽  
Ground State ◽  
Sufficient Conditions ◽  
Embedding Theorem ◽  
Compact Embedding ◽  
Concentration Compactness ◽  
Critical Nonlinearity ◽  
Concentration Compactness Principle ◽  
Laplace Equations ◽  
Compactness Principle

AbstractIn this paper, we use the rearrangement-free argument, in the spirit of the work by Li, Lu and Zhu [25], on the concentration-compactness principle on the Heisenberg group to establish a sharpened version of the singular Lions concentration-compactness principle for the Trudinger–Moser inequality in{\mathbb{R}^{n}}. Then we prove a compact embedding theorem, which states that{W^{1,n}(\mathbb{R}^{n})}is compactly embedded into{L^{p}(\mathbb{R}^{n},|x|^{-\beta}\,dx)}for{p\geq n}and{0<\beta<n}. As an application of the above results, we establish sufficient conditions for the existence of ground state solutions to the followingn-Laplace equation with critical nonlinearity:($*$){}\left\{\begin{aligned} &\displaystyle{-}\operatorname{div}(|\nabla u|^{n-2}% \nabla u)+V(x)|u|^{n-2}u=\frac{f(x,u)}{|x|^{\beta}},\\ &\displaystyle u\in W^{1,n}(\mathbb{R}^{n}),\quad u\geq 0,\end{aligned}\right.where{V(x)\geq c_{0}}for some positive constant{c_{0}}and{f(x,t)}behaves like{\exp(\alpha|t|^{\frac{n}{n-1}})}as{t\rightarrow+\infty}. This work improves substantially related results found in the literature.

scholarly journals EXISTENCE OF STEADY STATES FOR THE MAXWELL–SCHRÖDINGER–POISSON SYSTEM: EXPLORING THE APPLICABILITY OF THE CONCENTRATION–COMPACTNESS PRINCIPLE

2013 ◽  
Vol 23 (10) ◽  
pp. 1915-1938 ◽  
Author(s):  
I. CATTO ◽  
J. DOLBEAULT ◽  
O. SÁNCHEZ ◽  
J. SOLER
Keyword(s):  
Steady States ◽  
Concentration Compactness ◽  
Poisson System ◽  
Open Problems ◽  
Compactness Method ◽  
System A ◽  
Concentration Compactness Principle ◽  
Compactness Principle ◽  
Schrödinger System

This paper reviews recent results and open problems concerning the existence of steady states to the Maxwell–Schrödinger system. A combination of tools, proofs and results are presented in the framework of the concentration–compactness method.

scholarly journals Existence of ground state for coupled system of biharmonic Schrödinger equations

2021 ◽  
Vol 7 (3) ◽  
pp. 3719-3730
Author(s):  
Yanhua Wang ◽  
◽  
Min Liu ◽  
Gongming Wei ◽  
Keyword(s):  
Ground State ◽  
Coupling Parameter ◽  
Coupled System ◽  
Ground State Solution ◽  
Nehari Manifold ◽  
State Solution ◽  
Schrodinger Equations ◽  
Concentration Compactness ◽  
Schrödinger Equations ◽  
Whole Space

<abstract><p>In this paper we consider the following system of coupled biharmonic Schrödinger equations</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \ \left\{ \begin{aligned} \Delta^{2}u+\lambda_{1}u = u^{3}+\beta u v^{2}, \\ \Delta^{2}v+\lambda_{2}v = v^{3}+\beta u^{2}v, \end{aligned} \right. \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>where $ (u, v)\in H^{2}({\mathbb{R}}^{N})\times H^2(\mathbb R^N) $, $ 1\leq N\leq7 $, $ \lambda_{i} &gt; 0 (i = 1, 2) $ and $ \beta $ denotes a real coupling parameter. By Nehari manifold method and concentration compactness theorem, we prove the existence of ground state solution for the coupled system of Schrödinger equations. Previous results on ground state solutions are obtained in radially symmetric Sobolev space $ H_r^2(\mathbb R^N)\times H_r^2(\mathbb R^N) $. When $ \beta $ satisfies some conditions, we prove the existence of ground state solution in the whole space $ H^2(\mathbb R^N)\times H^2(\mathbb R^N) $.</p></abstract>

Sharpened Adams Inequality and Ground State Solutions to the Bi-Laplacian Equation in ℝ4

2018 ◽  
Vol 18 (3) ◽  
pp. 429-452 ◽  
Author(s):  
Lu Chen ◽  
Jungang Li ◽  
Guozhen Lu ◽  
Caifeng Zhang
Keyword(s):  
Ground State ◽  
Compact Embedding ◽  
Concentration Compactness ◽  
Critical Nonlinearity ◽  
Ground State Solutions ◽  
Laplacian Equation ◽  
Concentration Compactness Principle ◽  
Formula Group ◽  
Compactness Principle ◽  
Sharp Concentration

AbstractIn this paper, we establish a sharp concentration-compactness principle associated with the singular Adams inequality on the second-order Sobolev spaces in{\mathbb{R}^{4}}. We also give a new Sobolev compact embedding which states{W^{2,2}(\mathbb{R}^{4})}is compactly embedded into{L^{p}(\mathbb{R}^{4},|x|^{-\beta}\,dx)}for{p\geq 2}and{0<\beta<4}. As applications, we establish the existence of ground state solutions to the following bi-Laplacian equation with critical nonlinearity:\displaystyle\Delta^{2}u+V(x)u=\frac{f(x,u)}{|x|^{\beta}}\quad\mbox{in }% \mathbb{R}^{4},where{V(x)}has a positive lower bound and{f(x,t)}behaves like{\exp(\alpha|t|^{2})}as{t\to+\infty}. In the case{\beta=0}, because of the loss of Sobolev compact embedding, we use the principle of symmetric criticality to obtain the existence of ground state solutions by assuming{f(x,t)}and{V(x)}are radial with respect toxand{f(x,t)=o(t)}as{t\rightarrow 0}.

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