Ground states for asymptotically periodic Schrödinger-Poisson systems with critical growth

2014 ◽  
Vol 12 (10) ◽  
Author(s):  
Hui Zhang ◽  
Junxiang Xu ◽  
Fubao Zhang ◽  
Miao Du
Keyword(s):  
Ground States ◽  
Nehari Manifold ◽  
Critical Growth ◽  
Concentration Compactness ◽  
Concentration Compactness Principle ◽  
Asymptotically Periodic ◽  
Compactness Principle

AbstractFor a class of asymptotically periodic Schrödinger-Poisson systems with critical growth, the existence of ground states is established. The proof is based on the method of Nehari manifold and concentration compactness principle.

EXISTENCE AND MULTIPLICITY OF NONTRIVIAL SOLUTIONS FOR DEGENERATE HEMIVARIATIONAL INEQUALITIES INVOLVING LERAY–LIONS TYPE OPERATOR WITH CRITICAL GROWTH

2013 ◽  
Vol 11 (01) ◽  
pp. 1350007
Author(s):  
KAIMIN TENG
Keyword(s):  
Critical Growth ◽  
Type Operator ◽  
Hemivariational Inequalities ◽  
Concentration Compactness ◽  
Nontrivial Solutions ◽  
Existence And Multiplicity ◽  
Concentration Compactness Principle ◽  
Nonsmooth Critical Point ◽  
Critical Point Theorems ◽  
Compactness Principle

In this paper, we investigate a hemivariational inequality involving Leray–Lions type operator with critical growth. Some existence and multiple results are obtained through using the concentration compactness principle of P. L. Lions and some nonsmooth critical point theorems.

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 EXISTENCE OF SOLUTIONS FOR CRITICAL SYSTEMS WITH VARIABLE EXPONENTS

2018 ◽  
Vol 23 (4) ◽  
pp. 596-610 ◽  
Author(s):  
Hadjira Lalilia ◽  
Saadia Tas ◽  
Ali Djellit
Keyword(s):  
Existence Result ◽  
Existence Of Solutions ◽  
Elliptic Systems ◽  
Growth Conditions ◽  
Critical Growth ◽  
Concentration Compactness ◽  
Variable Exponents ◽  
Critical Systems ◽  
Concentration Compactness Principle ◽  
Compactness Principle

In this work, we deal with elliptic systems under critical growth conditions on the nonlinearities. Using a variant of concentration-compactness principle, we prove an existence result.

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.

scholarly journals Nonlinear perturbations of a periodic fractional Laplacian with supercritical growth

2021 ◽  
pp. 1-15
Author(s):  
Giovany M. Figueiredo ◽  
Sandra I. Moreira ◽  
Ricardo Ruviaro
Keyword(s):  
Variational Method ◽  
Positive Solutions ◽  
Fractional Laplacian ◽  
Concentration Compactness ◽  
Nonlinear Perturbations ◽  
Concentration Compactness Principle ◽  
Existence Of Positive Solutions ◽  
Asymptotically Periodic ◽  
Nonlinear Fractional Schrödinger Equation ◽  
Compactness Principle

Our main goal is to explore the existence of positive solutions for a class of nonlinear fractional Schrödinger equation involving supercritical growth given by $$ (- \Delta)^{\alpha} u + V(x)u=p(u),\quad x\in \mathbb{R^N},\ N \geq 1. $$ We analyze two types of problems, with $V$ being periodic and asymptotically periodic; for this we use a variational method, a truncation argument and a concentration compactness principle.

scholarly journals Multiple solutions for critical Choquard-Kirchhoff type equations

2020 ◽  
Vol 10 (1) ◽  
pp. 400-419 ◽  
Author(s):  
Sihua Liang ◽  
Patrizia Pucci ◽  
Binlin Zhang
Keyword(s):  
Critical Exponent ◽  
Critical Exponents ◽  
Sobolev Inequality ◽  
Multiple Solutions ◽  
Concentration Compactness ◽  
Positive Real ◽  
Multiplicity Results ◽  
Kirchhoff Type ◽  
Concentration Compactness Principle ◽  
Compactness Principle

Abstract In this article, we investigate multiplicity results for Choquard-Kirchhoff type equations, with Hardy-Littlewood-Sobolev critical exponents, $$\begin{array}{} \displaystyle -\left(a + b\int\limits_{\mathbb{R}^N} |\nabla u|^2 dx\right){\it\Delta} u = \alpha k(x)|u|^{q-2}u + \beta\left(\,\,\displaystyle\int\limits_{\mathbb{R}^N}\frac{|u(y)|^{2^*_{\mu}}}{|x-y|^{\mu}}dy\right)|u|^{2^*_{\mu}-2}u, \quad x \in \mathbb{R}^N, \end{array}$$ where a > 0, b ≥ 0, 0 < μ < N, N ≥ 3, α and β are positive real parameters, $\begin{array}{} 2^*_{\mu} = (2N-\mu)/(N-2) \end{array}$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality, k ∈ Lr(ℝN), with r = 2∗/(2∗ − q) if 1 < q < 2* and r = ∞ if q ≥ 2∗. According to the different range of q, we discuss the multiplicity of solutions to the above equation, using variational methods under suitable conditions. In order to overcome the lack of compactness, we appeal to the concentration compactness principle in the Choquard-type setting.

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