Concentration-compactness Principle Research Articles

Existence of Ground State Solutions for Fractional Schrödinger–Poisson Systems with Doubly Critical Growth

This paper considers a class of fractional Schrodinger–Poisson type systems with doubly critical growth $$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^su+V(x)u-\phi |u|^{2^*_s-3}u=K(x)|u|^{2^*_s-2}u,&{} \text{ in } {\mathbb {R}}^3,\\ (-\Delta )^s\phi =|u|^{2^*_s-1},&{} \text{ in } {\mathbb {R}}^3, \end{array}\right. \end{aligned}$$ where $$s\in (3/4,1)$$ , $$2^*_s=\frac{6}{3-2s}$$ , $$V\in L^{\frac{3}{2s}}({\mathbb {R}}^{3})$$ , $$K\in L^{\infty }({\mathbb {R}}^{3})$$ . By applying the concentration-compactness principle and variational method, the existence of ground state solutions to the systems is derived.

Dirac–Coulomb operators with general charge distribution II. The lowest eigenvalue

Consider the Coulomb potential $-\mu\ast|x|^{-1}$ generated by a non-negative finite measure $\mu$. It is well known that the lowest eigenvalue of the corresponding Schr\"odinger operator $-\Delta/2-\mu\ast|x|^{-1}$ is minimized, at fixed mass $\mu(\mathbb{R}^3)=\nu$, when $\mu$ is proportional to a delta. In this paper we investigate the conjecture that the same holds for the Dirac operator $-i\alpha\cdot\nabla+\beta-\mu\ast|x|^{-1}$. In a previous work on the subject we proved that this operator is self-adjoint when $\mu$ has no atom of mass larger than or equal to 1, and that its eigenvalues are given by min-max formulas. Here we consider the critical mass $\nu_1$, below which the lowest eigenvalue does not dive into the lower continuum spectrum for all $\mu\geq0$ with $\mu(\mathbb{R}^3)<\nu_1$. We first show that $\nu_1$ is related to the best constant in a new scaling-invariant Hardy-type inequality. Our main result is that for all $0\leq\nu<\nu_1$, there exists an optimal measure $\mu\geq0$ giving the lowest possible eigenvalue at fixed mass $\mu(\mathbb{R}^3)=\nu$, which concentrates on a compact set of Lebesgue measure zero. The last property is shown using a new unique continuation principle for Dirac operators. The existence proof is based on the concentration-compactness principle.

Nonlinear dispersive equations: classical and new frameworks

The purpose of this manuscript associated to the Golden Jubilee of the IME-USP is to present selected material from the author’s scientific contribution dealing with nonlinear phenomena of dispersive type. That study has been a source for modern research in the dynamic of traveling wave solutions of different type and it has been disseminated through publication of scientific articles and/or books. Here, we provide the reader with information about current research in stability theory for nonlinear dispersive equations and possible developments. Also, I hope it inspires future developments in this important and fascinating subject. In this manuscript we consider the following topics: stability theory of solitary waves and the applicability of the concentration–compactness principle, the existence and orbital (in)stability of periodic traveling wave solutions for nonlinear dispersive models, nonlinear Schrodinger and Korteweg–de Vries models on star-shaped metric graphs. The use of tools of the theory of spaces of Hilbert, the spectral theory for unbounded self-adjoint operators, Sturm–Liouville’s theory, variational methods, analytic perturbation theory of operators, and the extension theory of symmetric operators are pieces fundamental in our study. The methods presented in this manuscript have prospect for the study of the dynamic of solutions for nonlinear evolution equations around of different traveling waves profiles which may appear in non-standard environments such as star-shaped metric graphs.

The concentration-compactness principles for W s,p(·,·)(ℝ N ) and application

Abstract We obtain a critical imbedding and then, concentration-compactness principles for fractional Sobolev spaces with variable exponents. As an application of these results, we obtain the existence of many solutions for a class of critical nonlocal problems with variable exponents, which is even new for constant exponent case.

Multiplicity results for (p,\xa0q)-Laplacian equations with critical exponent in {mathbb {R}}^N and negative energy

We prove existence results in all of {mathbb {R}}^N for an elliptic problem of (p, q)-Laplacian type involving a critical term, nonnegative weights and a positive parameter lambda . In particular, under suitable conditions on the exponents of the nonlinearity, we prove existence of infinitely many weak solutions with negative energy when lambda belongs to a certain interval. Our proofs use variational methods and the concentration compactness principle. Towards this aim we give a detailed proof of tight convergence of a suitable sequence.

Existence and multiplicity results for critical anisotropic Kirchhoff-type problems with nonlocal nonlinearities

In this paper, we are interested in existence and multiplicity of solutions to anisotropic elliptic equations of Kirchhoff-type given by where Ω is a smooth bounded domain of and are parameters, where is the critical exponent and . The function is only continuous with and can change its sign, is a function with subcritical growth and Under appropriate assumptions on f, applying an adequate truncation argument on M and the Concentration Compactness-Principle for the anisotropic operator [1], we obtain the existence of a nontrivial solution for by the Mountain Pass Theorem [2] and multiplicity of solutions by Krasnoselskii's genus and the symmetric Mountain Pass Lemma [3]. As a model case for M and f, we can consider or with and with and appropriate

Normalized ground states for general pseudo-relativistic Schrödinger equations

ABSTRACT In this paper, we consider the pseudo-relativistic type Schrödinger equations with general nonlinearities. By studying the related constrained minimization problems, we obtain the existence of ground states via applying the concentration-compactness principle. Then some properties of the ground states have been discussed, including regularity, symmetry and etc. Furthermore, we prove that the set of minimizers is a stable set for the initial value problem of the equations, that is, a solution whose initial data is near the set will remain near it for all time.

Існування періодичних розв’язків в системі нелінійних осциляторів зі степеневими потенціалами на двовимірній ґратці

This article is devoted to the study of an infinite-dimensional Hamiltonian system, which describes an infinite system of linearly coupled nonlinear oscillators on a two-dimensional lattice. This system is a counteble system of ordinary differential equations. It is convenient to consider this system as a differential-operator equation in Hilbert space of real two-way sequences. The problem of existence of periodic solutions for such systems with power potential is considered. The main conditions for the existence of these solutions are the spatial periodicity of the coefficients of the linear interaction of oscillators and the positivity of this operator. This article shows that periodic solutions can be constructed using the constained minimization method. For this, a functional is constructed whose critical points are the desired periodic solutions. This functional is represented as the difference between the quadratic and non-quadratic parts. Next, we consider the problem of constrained minimization of the quadratic part of the functional under the condition that the non-quadratic part is constant. Using the Lagrange multiplier method, it was found that the periodic solutions of this system linearly depend on the solutions of the considered problem of constrained minimization, in particular, the coefficient of linear dependence is expressed in terms of the Lagrange multiplier. Using a discrete version of the concentration compactness principle, it is proved that the problem of constrained minimization under consideration has a solution, and therefore, there are periodic solutions of the original system. In particular, it is shown that for an arbitrary minimizing sequence of the quadratic part of the constructed functional, the possibility of «concentration» of the concentration compactness principle is satisfies, which allowed to prove the boundedness of this sequence. Moreover, it is proved that for sufficiently large values of periods these solutions are not constant

Extremal problems of Hardy-Littlewood-Sobolev inequalities on compact Riemannian manifolds

In this paper we study the existence of extremal problems for the Hardy-Littlewood-Sobolev inequalities on compact Riemannian manifolds without boundary via Concentration-Compactness principle.

Nontrivial solution for Schrödinger–Poisson equations involving the fractional Laplacian with critical exponent

This paper deals with a class of nonlocal Schrodinger equations with critical exponent $$\begin{aligned} \left\{ \begin{array}{llll} (-\Delta )^{s} u+V(x)u-K(x)\phi |u|^{2^*_s-3}u=f(x,u),&{}\quad \mathrm{in}\ \mathbb {R}^3,\\ (-\Delta )^s \phi =K(x)|u|^{2^*_s-1},&{}\quad \mathrm{in}\ \mathbb {R}^3. \end{array}\right. \end{aligned}$$ By employing the mountain pass theorem, concentration-compactness principle and approximation method, the existence of nontrivial solution is obtained under appropriate assumptions on V, K and f.

Generalized Concentration-Compactness Principles for Variable Exponent Lebesgue Spaces with Asymptotic Analysis of Low Energy Extremals

In this paper, we prove two generalized concentration-compactness principles for variable exponent Lebesgue spaces and as an application study the asymptotic behaviour of low energy extremals.

Infinitely Many Solutions to a Class of Fractional Elliptic System with Critical Sobolev Exponents in RN

The paper focuses on a class of fractional elliptic system with critical Sobolev exponents, where there is no compact embedding under proper assumptions on potential functions. The proof of the existence results mainly relies on concentration-compactness principle of fractional Sobolev space and genus theory.

Existence and concentration of ground state solutions for doubly critical Schrödinger–Poisson-type systems

In this paper, we are concerned with the existence and concentration of ground state solutions for the following nonlinear Schrodinger–Poisson-type system with doubly critical growth $$\begin{aligned} {\left\{ \begin{array}{ll} -\varepsilon ^2\Delta u+V(x)u-\phi |u|^3u=|u|^4u+f(u),&{}\mathrm{in}\; \mathbb {R}^3,\\ -\varepsilon ^2\Delta \phi =|u|^5,&{}\mathrm{in}\; \mathbb {R}^3, \end{array}\right. } \end{aligned}$$ where $$\varepsilon >0$$ is a small parameter. By employing the concentration-compactness principle and mountain pass theorem, we prove the existence of positive ground state solutions $$v_\varepsilon $$ with exponential decay at infinity for $$\varepsilon $$ sufficiently small under some suitable assumptions on the potential V and nonlinearity f. Moreover, as $$\varepsilon \rightarrow 0^+$$ , $$v_\varepsilon $$ concentrates around a global minimum point of V.

Concentration-compactness principle of singular Trudinger–Moser inequality involving N-Finsler–Laplacian operator

In this paper, suppose [Formula: see text] be a convex function of class [Formula: see text] which is even and positively homogeneous of degree 1. We establish the Lions type concentration-compactness principle of singular Trudinger–Moser Inequalities involving [Formula: see text]-Finsler–Laplacian operator. Let [Formula: see text] be a smooth bounded domain. [Formula: see text] be a sequence such that anisotropic Dirichlet norm[Formula: see text], [Formula: see text] weakly in [Formula: see text]. Denote [Formula: see text] Then we have [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text] is the volume of a unit Wulff ball. This conclusion fails if [Formula: see text]. Furthermore, we also obtain the corresponding concentration-compactness principle in the entire Euclidean space [Formula: see text].

Existence of solutions for a critical (p 1(x), … , pn(x))-Kirchhoff-type potential systems

ABSTRACT In this paper, by using the concentration–compactness principle of Lions for variable exponent spaces and the Mountain Pass Theorem, we obtain the existence of nontrivial solutions for a class of -Kirchhoff-type potential systems type with critical exponent.

Singular Supercritical Trudinger-Moser Inequalities and the Existence of Extremals

In this paper, we present the singular supercritical Trudinger-Moser inequalities on the unit ball B in ℝn, where n ≥ 2. More precisely, we show that for any given α > 0 and 0 < t < n, then the following two inequalities hold for ∀ u ∈ W0,1, (B), $$\mathop {\sup }\limits_{\int_B {{{\left| {\nabla u} \right|}^n}dx \le 1} } \,\int_B {{{\exp \left( {\left( {{\alpha _{n,t}} + {{\left| x \right|}^\alpha }} \right){{\left| u \right|}^{{n \over {n - 1}}}}} \right)} \over {{{\left| x \right|}^t}}}} dx < \infty $$ and $$\mathop {\sup }\limits_{\int_B {{{\left| {\nabla u} \right|}^n}dx \le 1} } \,\int_B {{{\exp \left( {{\alpha _{n,t}} + {{\left| u \right|}^{^{{n \over {n - 1}}} + {{\left| x \right|}^\alpha }}}} \right)} \over {{{\left| x \right|}^t}}}} dx < \infty .$$ We also consider the problem of the sharpness of the constant αn,t. Furthermore, by employing the method of estimating the lower bound and using the concentration-compactness principle, we establish the existence of extremals. These results extend the known results when t = 0 to the singular version for 0 < t < n.

Existence result for variable exponents elliptic system with lack of compactness

In this article, we prove the existence of a solution for nonlinear system governed by -Laplacian operators by using the concentration-compactness Principle for recovering the lack of compactness and some variable techniques.

An existence result for [formula omitted]-Laplace equations involving sandwich-type and critical growth

We investigate the existence of a nontrivial nonnegative solution to (p,q)-Laplace equations involving two nonlinear terms, one grows as s with q<s<p and the other possibly has critical growth. This interesting case cannot be appeared in single m-Laplace equations and has not been studied under even the subcritical growth. Our argument is based on the concentration–compactness principle by P.L. Lions and the Ekeland variational principle.

Existence and concentration of positive ground states for Schrodinger-Poisson equations with competing potential functions

This article concerns the Schrodinger-Poisson equation $$\displaylines{ -\varepsilon^2\Delta u+V(x)u+K(x)\phi u=P(x)|u|^{p-1}u+Q(x) |u|^{q-1}u,\quad x\in\mathbb{R}^3,\cr -\varepsilon^2\Delta \phi=K(x)u^2,\quad x\in\mathbb{R}^3, }$$ where \(3&lt;q&lt;p&lt;5=2^{\ast}-1\). We prove that for all \(\varepsilon&gt;0\), the equation has a ground state solution. The methods used here are based on the Nehari manifold and the concentration-compactness principle. Furthermore, for \varepsilon&gt;0 small, these ground states concentrate at a global minimum point of the least energy function.&#x0D; For more information see https://ejde.math.txstate.edu/Volumes/2020/78/abstr.html

Existence of ground state for fractional Kirchhoff equation with L^{2} critical exponents

In this paper, we consider a class of fractional Kirchhoff equations with L^{2} critical exponents. By using the scaling technique and concentration-compactness principle we obtain the existence and nonexistence of ground state for fractional Kirchhoff equation with L^{2} critical exponent.