Compactness Principle Research Articles

Ground state solution for fractional p-Choquard equations with upper critical exponent

In this paper, we investigate the following fractional p-Choquard equations with upper critical exponent involving two potentials:(−Δ)psu+V(x)|u|p−2u=[Iα⁎|u|ps,α⁎]|u|ps,α⁎−2u+K(x)f(x,u),x∈RN, where N⩾1, 0<s<1, 2⩽p<Ns, ps,α⁎=p(N+α)2(N−ps), Iα is the Riesz potential of order α∈(0,1), the potentials V,K:RN→R and nonlinearity f:RN×R→R are continuous functions satisfying some natural hypotheses. We show the existence of a ground state solution to above equations when V and f are periodic and asymptotically periodic in x, respectively, by using the idea of Nehari manifold technique and the Lions' concentration compactness principle.

Existence of solutions to a critical Kirchhoff equation with logarithmic perturbation

In this paper, the following Kirchhoff type elliptic equation − ( a + b ∫ Ω | ∇ u | 2 d x ) Δ u = λ | u | q − 2 u ln ⁡ | u | 2 + | u | 4 u , which involves a power type nonlinearity with critical Sobolev exponent and a logarithmic type perturbation, is investigated. The fact that the logarithmic term satisfies neither the monotonicity condition nor the Ambrosetti-Rabinowitz condition brings some additional difficulties when one is looking for weak solutions to this problem. By the use of the Mountain Pass Lemma and the concentration compactness principle, it is proved that the problem admits a ground state solution for q ∈ ( 4 , 6 ) and for any 0 $ ]]> λ > 0 . It is worth pointing out that the analysis includes both the nondegenerate case (a>0) and the degenerate case (a = 0).

Stability of constrained solitary waves for the Ostrovsky–Vakhnenko model in the coastal zone

In this paper, we consider a nonlinear water wave model describing the evolution of the internal tides in the coastal zone. First, we prove the existence of minimizers for the corresponding energy functional on the L2-norm constrained manifold by the concentrated compactness principle, and thus we obtain a ground state traveling wave solution of the model. Moreover, we show that the solution has exponential decay in the far field. Finally, we show that the traveling wave solution is non-degenerate, and thus the spectral stability of the solution can be obtained by using the instability index theory.

The Existence Result for a p -Kirchhoff-Type Problem Involving Critical Sobolev Exponent

In this paper, by using the mountain pass theorem and the concentration compactness principle, we prove the existence of a positive solution for a p -Kirchhoff-type problem with critical Sobolev exponent.

Multiplicity results for generalized quasilinear critical Schrödinger equations in RN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^N$$\\end{document}

Multiplicity results are proved for solutions both with positive and negative energy, as well as nonexistence results, of a generalized quasilinear Schrödinger potential free equation in the entire RN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^N$$\\end{document} involving a nonlinearity which combines a power-type term at a critical level with a subcritical term, both with weights. The equation has been derived from models of several physical phenomena such as superfluid film in plasma physics as well as the self-channelling of a high-power ultra-short laser in matter. Proof techniques, also in the symmetric setting, are based on variational tools, including concentration compactness principles, to overcome lack of compactness, and the use of a change of variable in order to deal with a well defined functional.

BIONIC PRINCIPLES OF ARCHITECTURAL AND PLANNING ORGANIZATION OF URBAN UNDERGROUND SPACES

The article analyzes bionic principles that are appropriate for use in the formation of an architectural environment, and takes into account the specifics of designing underground spaces using optimal connections and regularities used in nature. Was formulated the means of applying bionics and biomorphic structures in the formation of interiors and the similarity of residential buildings in the animal world in the development of functional connections of underground spaces. The safety of life activities when developing underground spaces as housing studied for the discomfort of being in a closed environment. The locality of using underground layers under buildings and structures does not solve the problem of complex underground communication and solving the problems of critical environmental. In the course of studying the principles of formation of a harmonious safe environment by representatives of flora and fauna, bionic principles of architectural and planning organization of urban underground spaces divided into two groups: technical and aesthetic. &#x0D; Technical principles that ensure the optimal functioning of the structure of the underground object: the principle of compactness; effective use of the landscape consists, creation of an artificial landscape; use of external energy sources; the principle of modularity, mobility, transformation, universality. &#x0D; Aesthetic principles of shaping the underground space, its configuration and decoration: the principle of proportioning, aesthetic integrity of the underground space, compensation for the absence of flora; principles of formation of natural color architectural space; integral principles of object fractality. &#x0D; The principle of potential development is aimed at identifying possible prospects for development, it includes a program for further modeling of the form, creation of an ultra-modern space, interactive, virtual, adaptive.

Almost sharp weighted Sobolev trace inequalities in the unit ball under constraints

In this paper, we establish some improved weighted Sobolev trace inequalities [Formula: see text] under the zero higher order moments constraint via the concentration compactness principle, where [Formula: see text] is a defining function of [Formula: see text] and [Formula: see text]. This relates to the fractional (conformal) Laplacians and related problems in conformal geometry. We construct some test functions and show that the inequality is almost optimal when [Formula: see text].

Indestructibility of some compactness principles over models of [formula omitted

We show that PFA (Proper Forcing Axiom) implies that adding any number of Cohen subsets of ω will not add an ω2-Aronszajn tree or a weak ω1-Kurepa tree, and moreover no σ-centered forcing can add a weak ω1-Kurepa tree (a tree of height and size ω1 with at least ω2 cofinal branches). This partially answers an open problem whether ccc forcings can add ω2-Aronszajn trees or ω1-Kurepa trees (with ¬□ω1 in the latter case).We actually prove more: We show that a consequence of PFA, namely the guessing model principle, GMP, which is equivalent to the ineffable slender tree property, ISP, is preserved by adding any number of Cohen subsets of ω. And moreover, GMP implies that no σ-centered forcing can add a weak ω1-Kurepa tree (see Section 2.1 for definitions).For more generality, we study variations of the principle GMP at higher cardinals and the indestructibility consequences they entail, and as applications we answer a question of Mohammadpour about guessing models at weakly but not strongly inaccessible cardinals and show that there is a model in which there are no weak ℵω+1-Kurepa trees and no ℵω+2-Aronszajn trees.

Integral inequalities with an extended Poisson kernel and the existence of the extremals

Abstract In this article, we first apply the method of combining the interpolation theorem and weak-type estimate developed in Chen et al. to derive the Hardy-Littlewood-Sobolev inequality with an extended Poisson kernel. By using this inequality and weighted Hardy inequality, we further obtain the Stein-Weiss inequality with an extended Poisson kernel. For the extremal problem of the corresponding Stein-Weiss inequality, the presence of double-weighted exponents not being necessarily nonnegative makes it impossible to obtain the desired existence result through the usual technique of symmetrization and rearrangement. We then adopt the concentration compactness principle of double-weighted integral operator, which was first used by the authors in Chen et al. to overcome this difficulty and obtain the existence of the extremals. Finally, the regularity of the positive solution for integral system related with the extended kernel is also considered in this article. Our regularity result also avoids the nonnegativity condition of double-weighted exponents, which is a common assumption in dealing with the regularity of positive solutions of the double-weighted integral systems in the literatures.

Image Enhancement Model Based on Fractional Time-Delay and Diffusion Tensor

Image enhancement is one of the bases of image processing technology, which can enhance useful features and suppress useless information of images according to the specified task. In order to ensure coherent enhancement for images with oriented flow-like structures, we propose a nonlinear diffusion system model based on time-fractional delay. By combining the nonlinear isotropic diffusion equation with fractional time-delay regularization, we construct a structure tensor. Meanwhile, the introduction of source terms enhances the contrast of the image, making it effective for denoising images with high-level noise. Based on compactness principles, the existence of weak solutions for the model is proved by using the Galerkin method. In addition, various experimental results verify the enhancement ability of the proposed model.

Multiplicity of positive solutions for a Kirchhoff type problem without asymptotic conditions

In this paper, we are concerned with the multiplicity of positive solutions for the following Kirchhoff type problem \[ \begin{cases} -\bigg({\varepsilon}^2a+{\varepsilon}b\int_{\mathbb{R}^3} |\n u|^2dx\bigg)\Delta u+u=Q(x)|u|^{p-2}u, &amp; x\in\mathbb{R}^3,\\ u\in H^1\big(\mathbb{R}^3\big), \quad u&gt; 0, &amp; x\in\mathbb{R}^3, \end{cases} \] where $\varepsilon&gt; 0$ is a small parameter, $a,b&gt; 0$ are constants, $4&lt; p&lt; 6$, $Q$ is a nonnegative continuous potential and does not satisfy any asymptotic condition. Combining Nehari manifold and concentration compactness principle, we study how the shape of the graph of $Q(x)$ affects the number of positive solutions.

Breather solutions for a semilinear Klein-Gordon equation on a periodic metric graph

We consider the nonlinear Klein-Gordon equation∂t2u(x,t)−∂x2u(x,t)+αu(x,t)=±|u(x,t)|p−1u(x,t) on a periodic metric graph (necklace graph) for p>1 with Kirchhoff conditions at the vertices. Under suitable assumptions on the frequency we prove the existence and regularity of infinitely many spatially localized time-periodic solutions (breathers) by variational methods. Compared to previous results obtained via spatial dynamics and center manifold techniques our results provide existence for all values of α≥0 as well as multiplicity. Moreover, we deduce regularity properties of the solutions and show that they are weak solutions of the corresponding initial value problem. Our approach relies on the existence of critical points for indefinite functionals, the concentration compactness principle, and the proper set-up of a functional analytic framework. Compared to earlier work for breathers using variational techniques, a major improvement of embedding properties has been achieved. This allows in particular to avoid all restrictions on the exponent p>1 and to achieve higher regularity.

Sharp scattering for focusing intercritical NLS on high-dimensional waveguide manifolds

We study the focusing intercritical NLS NLS $$\begin{aligned} i\partial _t u+\Delta _{x,y}u=-|u|^\alpha u \end{aligned}$$ on the semiperiodic waveguide manifold $$\mathbb {R}^d_x\times \mathbb {T}_y$$ with $$d\ge 5$$ and $$\alpha \in (\frac{4}{d},\frac{4}{d-1})$$ . In the case $$d\le 4$$ , with the aid of the semivirial vanishing theory (Luo in Normalized ground states and threshold scattering for focusing NLS on Rd $$\times $$ T via semivirial-free geometry, 2022), the author was able to construct a sharp threshold, which being uniquely characterized by the ground state solutions, that sharply determines the bifurcation of global scattering and finite time blow-up solutions in dependence of the sign of the semivirial functional. As the derivative of the nonlinear potential is no longer Lipschitz in $$d\ge 5$$ and the underlying domain possesses an anisotropic nature, the proof in Luo (Normalized ground states and threshold scattering for focusing NLS on Rd $$\times $$ T via semivirial-free geometry, 2022), which makes use of the concentration compactness principle, can not be extended to higher dimensional models. In this paper, we exploit a well-tailored adaptation of the interaction Morawetz–Dodson–Murphy (IMDM) estimates, which were only known to be applicable on Euclidean spaces, into the waveguide setting, in order to prove that the large data scattering result formulated in Luo (Normalized ground states and threshold scattering for focusing NLS on Rd $$\times $$ T via semivirial-free geometry, 2022) continues to hold for all $$d\ge 5$$ . Together with Tzvetkov–Visciglia (Rev. Mat. Iberoam. 32:1163–1188, 2016) and the author (Luo in Normalized ground states and threshold scattering for focusing NLS on Rd $$\times $$ T via semivirial-free geometry, 2022), we thus give a complete characterization of the large data scattering for (NLS) in both defocusing and focusing case and in arbitrary dimension.

Compactness of Fixed Point Maps and the Ball-Marsden-Slemrod Conjecture

Given a parameter dependent fixed point equation $x = F(x,u)$, we derive an abstract compactness principle for the fixed point map $u \mapsto x^*(u)$ under the assumptions that (i) the fixed point equation can be solved by the contraction principle and (ii) the map $u \mapsto F(x,u)$ is compact for fixed $x$. This result is applied to infinite-dimensional, semi-linear control systems and their reachable sets. More precisely, we extend a non-controllability result of Ball, Marsden, and Slemrod [1] to semi-linear systems. First we consider $L^p$-controls, $p>1$. Subsequently we analyze the case $p=1$.

Normalized saddle solutions for a mass supercritical Choquard equation

We investigate the existence of saddle type normalized solutions for the nonlinear Choquard equation:{−Δu−λu=(Iα⁎F(u))F′(u), in RN∫RN|u|2dx=a2,u∈H1(RN). Here N≥1, a>0 is given in advance, Iα is the Riesz potential of order α∈(0,N) and the unknown parameter λ appears as a Lagrange multiplier. In a mass supercritical setting on F, we prove the existence of a couple (uaG,λaG)∈H1(RN)×R− of saddle solutions for any a>0 and for given finite Coxeter group G with its rank k≤N. Our method is to combine the concentration compactness principle with a minimax procedure in the saddle type symmetric subspace, which gives a variational framework of constructing normalized saddle solutions for the Choquard equation.

Nonlinear elliptic–parabolic problem involving p-Dirichlet-to-Neumann operator with critical exponent

Abstract We consider the nonlinear elliptic–parabolic boundary value problem involving the Dirichlet-to-Neumann operator of p-Laplace type at the critical Sobolev exponent. We first obtain the existence and asymptotic estimates of the global solution, and the sufficient conditions of finite time blowup of the solution by using the energy method. Second, we improve the regularity of solution by Moser-type iteration. Finally, we analyze the long-time asymptotic behavior of the global solution. Moreover, with the help of the concentration compactness principle, we present a precise description of the concentration phenomenon of the solution in the forward time infinity.

Existence of ground‐state solutions for p‐Choquard equations with singular potential and doubly critical exponents

AbstractWe are concerned with the following Choquard equation: where , , , , is the p‐Laplacian, is the Riesz potential, and F is the primitive of f which is of critical growth due to the Hardy–Littlewood–Sobolev inequality. Under different range of θ and almost necessary conditions on the nonlinearity f in the spirit of Berestycki–Lions‐type conditions, we divide this paper into three parts. By applying the refined Sobolev inequality with Morrey norm and the generalized version of the Lions‐type theorem, some existence results are established. It is worth noting that our method is not involving the concentration–compactness principle.

The Neumann problem for a class of generalized Kirchhoff-type potential systems

In this paper, we are concerned with the Neumann problem for a class of quasilinear stationary Kirchhoff-type potential systems, which involves general variable exponents elliptic operators with critical growth and real positive parameter. We show that the problem has at least one solution, which converges to zero in the norm of the space as the real positive parameter tends to infinity, via combining the truncation technique, variational method, and the concentration–compactness principle for variable exponent under suitable assumptions on the nonlinearities.

Normalized solutions to the Kirchhoff equation with a perturbation term

In this paper, we study the existence of solutions to the Kirchhoff equation \begin{equation*} - \Big ( {a + b\int_{{\mathbb{R}}^{3}} {|\nabla u{|^2}dx} } \Big ) \Delta u = \lambda u + |u{|^{p - 2}}u + \mu |u{|^{q - 2}}u~~{\rm in}~{\mathbb{R}}^{3}, \end{equation*} having prescribed mass $$ \int_{{\mathbb{R}}^{3}} {|u{|^2}dx} = c, $$ where $a,b > 0$, $\mu \in {\mathbb{R}}$, $2 < q < p < 6$. When $(p,q)$ belongs to a certain domain in ${{\mathbb{R}}^{2}}$, we prove the existence and nonexistence of normalized solutions by using constraint minimization and concentration compactness principle, our main results may be illustrated by the red areas and green areas shown in Figure 1. In particular, our results are closely related to the values of $\mu$ and prescribed mass $c$, and partially extend the results of Li et al. [10] and Soave [20].

Fractional Schrödinger–Poisson system with critical growth and potentials vanishing at infinity

AbstractIn this paper, we are concerned with the existence of positive solutions for a class of fractional Schrödinger–Poisson system with critical nonlinearity and multiple competing potentials, which may decay and vanish at infinity. Under some local conditions, we show the existence and concentration of positive solutions by using the modified penalization method and concentration–compactness principle.