Global Compactness Research Articles

Bound states for fractional Kirchhoff equations with critical growth

This paper deals with the following fractional Kirchhoff problems: ( ε 2 s a + ε 4 s − N b ∫ R N | ( − Δ ) s 2 u | 2 d x ) ( − Δ ) s u + V ( x ) u = | u | 2 s ∗ − 2 u , in R N , where a, b>0, s ∈ ( 0 , 1 ) , 2s<N<4s, ε is a positive parameter, 2 s ∗ = 2 N N − 2 s is the critical Sobolev exponent, V ( x ) ∈ L N 2 s ( R N ) and V ( x ) vanishes at some points in R N . In virtue of a global compactness result and Ljusternik–Schnirelman theory, we succeed in proving the multiplicity of bound states.

Palaeohistology of Macrospondylus bollensis (Crocodylomorpha: Thalattosuchia: Teleosauroidea) from the Posidonienschiefer Formation (Toarcian) of Germany, with insights into life history and ecology.

The Posidonienschiefer Formation of southern Germany has yielded an array of incredible fossil vertebrates. One of the best represented clades therein is Teleosauroidea, a successful thalattosuchian crocodylomorph group that dominated the coastlines. The most abundant teleosauroid, Macrospondylus bollensis, is known from a wide range of body sizes, making it an ideal taxon for histological and ontogenetic investigations. Previous studies examining thalattosuchian histology provide a basic understanding of bone microstructure in teleosauroids, but lack the taxonomic, stratigraphic, and ontogenetic control required to understand growth and palaeobiology within a species. Here, we examine the bone microstructure of three femora and one tibia from three different-sized M. bollensis individuals. We also perform bone compactness analyses to evaluate for ontogenetic and ecological variation. Our results suggests that (1) the smallest specimen was a young, skeletally immature individual with well-vascularized-parallel-fibered bone and limited remodeling in the midshaft periosteal cortex; (2) the intermediate specimen was skeletally immature at death, with vascularized parallel-fibered bone tissue interrupted by at least 10 LAGs, but no clear external fundamental system (EFS), and rather extensive inner cortical bone remodeling; and (3) the largest specimen was skeletally mature, with parallel-fibered bone tissue interrupted by numerous LAGs, a well-developed EFS, and extensive remodeling in the deep cortex. Macrospondylus bollensis grew relatively regularly until reaching adult size, and global bone compactness values fall within the range reported for modern crocodylians. The lifestyle inference models used suggest that M. bollensis was well adapted for an aquatic environment but also retained some ability to move on land. Finally, both larger specimens display a peculiar, localized area of disorganized bone tissue interpreted as pathological.

A Global compactness result and multiplicity of solutions for a class of critical exponent problems in the hyperbolic space

A Global compactness result and multiplicity of solutions for a class of critical exponent problems in the hyperbolic space

Multiple bound states for a class of fractional critical Schrödinger–Poisson systems with critical frequency

In this paper we study the fractional Schrödinger–Poisson system ε2s(−Δ)su+V(x)u=ϕ|u|2s*−3u+|u|2s*−2u,ε2s(−Δ)sϕ=|u|2s*−1,x∈R3, where s ∈ (0, 1), ɛ &amp;gt; 0 is a small parameter, 2s*=63−2s is the critical Sobolev exponent and V∈L32s(R3) is a nonnegative function which may be zero in some regions of R3, e.g., it is of the critical frequency case. By virtue of a new global compactness lemma, and the Lusternik–Schnirelmann category theory, we relate the number of bound state solutions with the topology of the zero set where V attains its minimum for small values of ɛ.

Multiple high energy solutions of a nonlinear Hardy–Sobolev critical elliptic equation arising in astrophysics

In this article, we study the existence and multiplicity of high energy solutions to the problem proposed as a model for the dynamics of galaxies: −Δu+V(x)u=|u|2∗−2u|y|,x=(y,z)∈Rm×Rn−m,where n>4, 2≤m<n, 2∗≔2(n−1)n−2 and potential function V(x):Rn→R. Benefiting from a global compactness result, we show that there exist at least two positive high energy solutions. Our proofs are based on barycenter function, quantitative deformation lemma and Brouwer degree theory.

Spatial Chromosome Organization and Adaptation of Escherichia coli under Heat Stress.

The spatial organization of bacterial chromosomes is crucial for cellular functions. It remains unclear how bacterial chromosomes adapt to high-temperature stress. This study delves into the 3D genome architecture and transcriptomic responses of Escherichia coli under heat-stress conditions to unravel the intricate interplay between the chromosome structure and environmental cues. By examining the role of macrodomains, chromosome interaction domains (CIDs), and nucleoid-associated proteins (NAPs), this work unveils the dynamic changes in chromosome conformation and gene expression patterns induced by high-temperature stress. It was observed that, under heat stress, the short-range interaction frequency of the chromosomes decreased, while the long-range interaction frequency of the Ter macrodomain increased. Furthermore, two metrics, namely, Global Compactness (GC) and Local Compactness (LC), were devised to measure and compare the compactness of the chromosomes based on their 3D structure models. The findings in this work shed light on the molecular mechanisms underlying thermal adaptation and chromosomal organization in bacterial cells, offering valuable insights into the complex inter-relationships between environmental stimuli and genomic responses.

Structure elucidation of a multi-modular recombinant endoglucanase, AtGH9C-CBM3A-CBM3B from Acetivibrio thermocellus ATCC 27405 and its substrate binding analysis

Cellulases from GH9 family show endo-, exo- or processive endocellulase activity, but the reason behind the variation is unclear. A GH9 recombinant endoglucanase, AtGH9C-CBM3A-CBM3B from Acetivibrio thermocellus was structurally characterized for conformation, binding and dynamics assessment. Modeled AtGH9C-CBM3A-CBM3B depicted (α/α)6-barrel structure with Asp98, Asp101 and Glu489 acting as catalytic triad. CD results revealed 25.2 % α-helix, 18.4 % β-sheet and rest 56.4 % of random coils, corroborating with predictions from PSIPRED and SOPMA. MD simulation of AtGH9C-CBM3A-CBM3B bound cellotetraose showed structural stability and global compactness with lowered RMSD values (1.5 nm) as compared with only AtGH9C-CBM3A-CBM3B (1.8 nm) for 200 ns. Higher fluctuation in RMSF values in far-positioned CBM3B pointed to its redundancy in substrate binding. Docking studies showed maximum binding with cellotetraose (ΔG = −5.05 kcal/mol), with reduced affinity towards ligands with degree of polymerization (DP) lower (DP < 4) or higher than 4 (DP > 4). Processivity index displayed the enzyme to be processive with loop 3 (342–379 aa) possibly blocking the non-reducing end of cellulose chain, resulting in cellotetraose release. SAXS analysis of AtGH9C-CBM3A-CBM3B at 5 mg/mL displayed monodispersed state with fist-and-elbow shape in solution. Negative zeta potential of −24 mV at 5 mg/mL indicated stability and free from aggregation.

Multiple high‐energy solutions for an elliptic system with critical Hardy–Sobolev nonlinearity

This paper discusses the existence of multiple high‐energy solutions for a ‐Laplacian system involving critical Hardy‐Sobolev nonlinearity in . Considering that the “double” lack of compactness in the system is caused by the unboundedness of and the presence of the critical Hardy–Sobolev exponent, we demonstrate the version to of Struwe's classical global compactness result for double ‐Laplace operator. In virtue of the quantitative deformation lemma, a barycenter function, and the Brouwer degree theory, the existence of multiple high‐energy solutions is established. The results of this paper extend and complement the recent work.

Diving dinosaurs? Caveats on the use of bone compactness and pFDA for inferring lifestyle.

The lifestyle of spinosaurid dinosaurs has been a topic of lively debate ever since the unveiling of important new skeletal parts for Spinosaurus aegyptiacus in 2014 and 2020. Disparate lifestyles for this taxon have been proposed in the literature; some have argued that it was semiaquatic to varying degrees, hunting fish from the margins of water bodies, or perhaps while wading or swimming on the surface; others suggest that it was a fully aquatic underwater pursuit predator. The various proposals are based on equally disparate lines of evidence. A recent study by Fabbri and coworkers sought to resolve this matter by applying the statistical method of phylogenetic flexible discriminant analysis to femur and rib bone diameters and a bone microanatomy metric called global bone compactness. From their statistical analyses of datasets based on a wide range of extant and extinct taxa, they concluded that two spinosaurid dinosaurs (S. aegyptiacus, Baryonyx walkeri) were fully submerged "subaqueous foragers," whereas a third spinosaurid (Suchomimus tenerensis) remained a terrestrial predator. We performed a thorough reexamination of the datasets, analyses, and methodological assumptions on which those conclusions were based, which reveals substantial problems in each of these areas. In the datasets of exemplar taxa, we found unsupported categorization of taxon lifestyle, inconsistent inclusion and exclusion of taxa, and inappropriate choice of taxa and independent variables. We also explored the effects of uncontrolled sources of variation in estimates of bone compactness that arise from biological factors and measurement error. We found that the ability to draw quantitative conclusions is limited when taxa are represented by single data points with potentially large intrinsic variability. The results of our analysis of the statistical method show that it has low accuracy when applied to these datasets and that the data distributions do not meet fundamental assumptions of the method. These findings not only invalidate the conclusions of the particular analysis of Fabbri et al. but also have important implications for future quantitative uses of bone compactness and discriminant analysis in paleontology.

A variant prescribed curvature flow on closed surfaces with negative Euler characteristic

On a closed Riemannian surface (M,{bar{g}}) with negative Euler characteristic, we study the problem of finding conformal metrics with prescribed volume A>0 and the property that their Gauss curvatures f_lambda = f + lambda are given as the sum of a prescribed function f in C^infty (M) and an additive constant lambda . Our main tool in this study is a new variant of the prescribed Gauss curvature flow, for which we establish local well-posedness and global compactness results. In contrast to previous work, our approach does not require any sign conditions on f. Moreover, we exhibit conditions under which the function f_lambda is sign changing and the standard prescribed Gauss curvature flow is not applicable.

Fractional nonhomogeneous system with Hardy-Littlewood-Sobolev critical nonlinearity

In this paper, we consider the following fractional elliptic system of Choquard type in where , N>2s, , is the upper critical exponent in the Hardy-Littlewood-Sobolev inequality and are nonnegative functionals in the dual space of . When , we establish the uniqueness of the solution to the above problem. On the other hand, when and are nonnegative functionals with , the multiplicity of solutions to the above problem is also shown. Moreover, we obtain the global compactness result by using (PS) decomposition.

Ground State Solutions of Fractional Choquard Problems with Critical Growth

In this article, we investigate a class of fractional Choquard equation with critical Sobolev exponent. By exploiting a monotonicity technique and global compactness lemma, the existence of ground state solutions for this equation is obtained. In addition, we demonstrate the existence of ground state solutions for the corresponding limit problem.

Ground states for nonlinear fractional Schrödinger–Poisson systems with general convolution nonlinearities

In this paper, we study the following fractional Schrödinger–Poisson system where is the Riesz potential, . By using a monotonicity trick and global compactness lemma, we obtain the existence of a ground state solution for the above system.

A global compactness result with applications to a nonlinear elliptic equation arising in astrophysics

In this paper, we study a nonlinear elliptic equation arising in astrophysics:(0.1)−Δu+V(x)u=|u|2⁎−2u|x′|+Q(x)|u|q−2u,x:=(x′,x″)∈Rm×Rn−m, where n≥3, 2≤m<n, 2⁎=2(n−1)n−2, 2<q<2nn−2 and V(x),Q(x)∈C(Rn). The partly singular term |u|2⁎−2u|x′| in (0.1) forces us to establish a refined Sobolev inequality involving partly weighted Morrey norm, which generalizes similar inequality obtained by G. Palatucci and A. Pisante (2014) [25]. Benefiting from the new inequality, we obtain a global compactness result and some existence result for (0.1), which extend the results obtained by Y. B. Deng et al. (2012) [12]. Our strategy turns out to be more concise because we avoid the use of Levy concentration functions.

Multiple solutions for a fractional Choquard problem with slightly subcritical exponents on bounded domains

This paper is devoted to study a fractional Choquard problem with slightly subcritical exponents on bounded domains. When the exponent of the convolution type nonlinearity tends to the fractional critical one in the sense of Hardy–Littlewood–Sobolev inequality, we obtain the existence of multiple positive solutions via Lusternik–Schnirelmann category and nonlocal global compactness. Moreover, we prove that the topology of the domain furnishes a lower bound for the number of positive solutions.

A global compactness result with applications to a Hardy-Sobolev critical elliptic system involving coupled perturbation terms

Abstract In this article, we study a Hardy-Sobolev critical elliptic system involving coupled perturbation terms: (0.1) − Δ u + V 1 ( x ) u = η 1 η 1 + η 2 ∣ u ∣ η 1 − 2 u ∣ v ∣ η 2 ∣ x ′ ∣ + α α + β Q ( x ) ∣ u ∣ α − 2 u ∣ v ∣ β , − Δ v + V 2 ( x ) v = η 2 η 1 + η 2 ∣ v ∣ η 2 − 2 v ∣ u ∣ η 1 ∣ x ′ ∣ + β α + β Q ( x ) ∣ v ∣ β − 2 v ∣ u ∣ α , \left\{\begin{array}{c}-\Delta u+{V}_{1}\left(x)u=\frac{{\eta }_{1}}{{\eta }_{1}+{\eta }_{2}}\frac{{| u| }^{{\eta }_{1}-2}u{| v| }^{{\eta }_{2}}}{| x^{\prime} | }+\frac{\alpha }{\alpha +\beta }Q\left(x)| u{| }^{\alpha -2}u| v{| }^{\beta },\\ -\Delta v+{V}_{2}\left(x)v=\frac{{\eta }_{2}}{{\eta }_{1}+{\eta }_{2}}\frac{{| v| }^{{\eta }_{2}-2}v{| u| }^{{\eta }_{1}}}{| x^{\prime} | }+\frac{\beta }{\alpha +\beta }Q\left(x){| v| }^{\beta -2}v{| u| }^{\alpha },\end{array}\right. where n ≥ 3 n\ge 3 , 2 ≤ m &lt; n 2\le m\lt n , x ≔ ( x ′ , x ″ ) ∈ R m × R n − m x:= \left(x^{\prime} ,{x}^{^{\prime\prime} })\in {{\mathbb{R}}}^{m}\times {{\mathbb{R}}}^{n-m} , η 1 , η 2 &gt; 1 {\eta }_{1},{\eta }_{2}\gt 1 , and η 1 + η 2 = 2 ( n − 1 ) n − 2 {\eta }_{1}+{\eta }_{2}=\frac{2\left(n-1)}{n-2} , α , β &gt; 1 \alpha ,\beta \gt 1 and α + β &lt; 2 n n − 2 \alpha +\beta \lt \frac{2n}{n-2} , and V 1 ( x ) , V 2 ( x ) , Q ( x ) ∈ C ( R n ) {V}_{1}\left(x),{V}_{2}\left(x),Q\left(x)\in C\left({{\mathbb{R}}}^{n}) . Observing that (0.1) is doubly coupled, we first develop two efficient tools (i.e., a refined Sobolev inequality and a variant of the “Vanishing” lemma). On the previous tools, we will establish a global compactness result (i.e., a complete description for the Palais-Smale sequences of the corresponding energy functional) and some existence result for (0.1) via variational method. Our strategy turns out to be very concise because we avoid the use of Levy concentration functions and truncation techniques.

Least energy sign-changing solutions for Schrödinger-Poisson systems with potential well

Abstract In this article, we investigate the existence of least energy sign-changing solutions for the following Schrödinger-Poisson system − Δ u + V ( x ) u + K ( x ) ϕ u = f ( u ) , x ∈ R 3 , − Δ ϕ = K ( x ) u 2 , x ∈ R 3 , \left\{\begin{array}{ll}-\Delta u+V\left(x)u+K\left(x)\phi u=f\left(u),\hspace{1.0em}&amp; x\in {{\mathbb{R}}}^{3},\\ -\Delta \phi =K\left(x){u}^{2},\hspace{1.0em}&amp; x\in {{\mathbb{R}}}^{3},\\ \hspace{1.0em}\end{array}\right. where the functions V ( x ) , K ( x ) V\left(x),K\left(x) have finite limits as ∣ x ∣ → ∞ | x| \to \infty satisfying some mild assumptions. By combining variational methods with the global compactness lemma, we obtain a least energy sign-changing solution with exactly two nodal domains, and its energy is strictly larger than twice that of least energy solutions.

Least energy sign-changing solutions for Kirchhoff-type problems with potential well

In this paper, we investigate the existence of least energy sign-changing solutions for the Kirchhoff-type problem −a+b∫R3|∇u|2dxΔu+V(x)u=f(u),x∈R3, where a, b &amp;gt; 0 are parameters, V∈C(R3,R), and f∈C(R,R). Under weaker assumptions on V and f, by using variational methods with the aid of a new version of global compactness lemma, we prove that this problem has a least energy sign-changing solution with exactly two nodal domains, and its energy is strictly larger than twice that of least energy solutions.

Ground states for Dirac equation with singular potential and asymptotically periodic condition

In this paper we study the existence and asymptotic analysis of ground states for nonlinear Dirac equation with singular potential. Under the asymptotically periodic condition, using variational tools from non-Nehari manifold method, we establish a global compactness result and we prove that the existence of ground state solution and the continuous dependence of ground state energy about parameter as well as the asymptotic convergence of solutions.

Existence and Multiplicity of Bound State Solutions to a Kirchhoff Type Equation with a General Nonlinearity

In this paper, we consider the following Kirchhoff type equation $$\begin{aligned} -\left( a+ b\int _{{\mathbb {R}}^3}|\nabla u|^2\right) \Delta {u} +V(x)u=f(u),\,\,x\in {\mathbb {R}}^3, \end{aligned}$$ where $$a,b>0$$ and $$f\in C({\mathbb {R}},{\mathbb {R}})$$ , and the potential $$V\in C^1({\mathbb {R}}^3,{\mathbb {R}})$$ is positive, bounded and satisfies suitable decay assumptions. By using a perturbation approach together with a new version of global compactness lemma of Kirchhoff type, we prove the existence and multiplicity of bound state solutions for the above problem with a general nonlinearity. We especially point out that neither the Ambrosetti-Rabinowitz condition with $$\mu >4$$ nor any monotonicity assumption on f is required. Moreover, the potential V may not be radially symmetry or coercive. As a prototype, the nonlinear term involves the power-type nonlinearity $$f(u) =|u|^{p-2}u$$ for $$p\in (2, 6)$$ .