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In this article, we study the following Kirchhoff equation with combined nonlinearities: −a+b∫R4∣∇u∣2dxΔu+λu=μ∣u∣q−2u+∣u∣2u,inR4,∫R4∣u∣2dx=c2,\left\{\begin{array}{l}-\left(a+b\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{4}}{| \nabla u| }^{2}{\rm{d}}x\right)\Delta u+\lambda u=\mu {| u| }^{q-2}u+{| u| }^{2}u,\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{4},\\ \mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{4}}{| u| }^{2}{\rm{d}}x={c}^{2},\end{array}\right. where a,b,c>0a,b,c\gt 0, μ,λ∈R\mu ,\lambda \in {\mathbb{R}}, 20b,c\gt 0 and μ∈R\mu \in {\mathbb{R}}, we prove some existence, nonexistence, and asymptotic behavior of the obtained normalized solutions. When μ>0\mu \gt 0:(i) for 2

This article is devoted to studying the existence of positive solutions to the following fractional Choquard equation: (−Δ)su+u=∫Ω∣u(y)∣p∣x−y∣N−αdy∣u∣p−2u+ε∫Ω∣u(y)∣2α,s*∣x−y∣N−αdy∣u∣2α,s*−2u,inΩ,u=0,onRN\Ω,\left\{\begin{array}{ll}{\left(-\Delta )}^{s}u+u=\left(\mathop{\displaystyle \int }\limits_{\Omega }\frac{{| u(y)| }^{p}}{{| x-y| }^{N-\alpha }}{\rm{d}}y\right){| u| }^{p-2}u+\varepsilon \left(\mathop{\displaystyle \int }\limits_{\Omega }\frac{{| u(y)| }^{{2}_{\alpha ,s}^{* }}}{{| x-y| }^{N-\alpha }}{\rm{d}}y\right){| u| }^{{2}_{\alpha ,s}^{* }-2}u,& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ u=0,& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}\backslash \Omega \right,\end{array}\right. where Ω\Omega is an exterior domain with smooth boundary ∂Ω≠∅\partial \Omega \ne \varnothing such that RN\Ω{{\mathbb{R}}}^{N}\backslash \Omega is bounded, N>2s,20\varepsilon \gt 0 is a parameter. We establish the limit profiles and uniqueness of positive radial ground-states for the limit equation without the critical exponent as α\alpha sufficiently close to NN. Then, combining variational method, barycentric functions, and Brouwer degree theory, we determine the existence of positive bound-state solutions provided that ε>0\varepsilon \gt 0 is sufficiently small.

Ubiquitination, a post-translational modification that attaches one or more ubiquitin (Ub) molecules to another protein, plays a crucial role in the phase-separation processes. Ubiquitination can modulate the formation of membrane-less organelles in two ways. First, a scaffold protein drives phase separation, and Ub is recruited to the condensates. Second, Ub actively phase-separates through the interactions with other proteins. Thus, the role of ubiquitination and the resulting polyUb chains ranges from bystanders to active participants in phase separation. Moreover, long polyUb chains may be the primary driving force for phase separation. We further discuss that the different roles can be determined by the lengths and linkages of polyUb chains which provide preorganized and multivalent binding platforms for other client proteins. Together, ubiquitination adds a new layer of regulation for the flow of material and information upon cellular compartmentalization of proteins.

In this article, we study the following Klein-Gordon-Maxwell system: −Δu−(2ω+ϕ)ϕu=g(u),inR3,Δϕ=(ω+ϕ)u2,inR3,\left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{l}-\Delta u-\left(2\omega +\phi )\phi u=g\left(u),\hspace{1.0em}{\rm{in}}\hspace{1em}{{\mathbb{R}}}^{3},\hspace{1.0em}\\ \Delta \phi =\left(\omega +\phi ){u}^{2},\hspace{1.0em}{\rm{in}}\hspace{1em}{{\mathbb{R}}}^{3},\hspace{1.0em}\end{array}\right. where ω\omega is a constant that stands for the phase; uu and ϕ\phi are unknowns and gg satisfies the Berestycki-Lions condition [Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal. 82 (1983), 313–345; Nonlinear scalar field equations. II. Existence of infinitelymany solutions, Arch. Rational Mech. Anal. 82 (1983), 347–375]. The Klein-Gordon-Maxwell system is a model describing solitary waves for the nonlinear Klein-Gordon equation interacting with an electromagnetic field. By using variational methods and some analysis techniques, the existence of positive solution and multiple solutions can be obtained. Moreover, we study the properties of decay estimates and asymptotic behavior for the positive solution.

We study the existence of solutions for the quasilinear Schrödinger equation with the critical exponent and steep potential well. By using a change of variables, the quasilinear equations are reduced to a semilinear one, whose associated functionals satisfy the geometric conditions of the Mountain Pass Theorem for suitable assumptions. The existence of a ground state solution is obtained, and its concentration behavior is also considered.

As radiation therapy is increasingly utilized in the treatment of cancer, neuropathic pain (NP) is a common radiotherapy-related adverse effect and has a significant impact on clinical outcomes negatively. However, despite an improved understanding of neuropathic pain management, pain is often undertreated in patients with cancer. Herein, we reported two cases with radiotherapy-related neuropathic pain (RRNP) who presented a positive reaction to acupuncture. Patient 1 (a 73-year-old woman) with gynecologic cancer complained of burning and electric shock-like pain in the lower limb after radiotherapy. With the accepted combination of acupuncture and drugs, the pain was alleviated completely in 8 weeks. Patient 2 (a 64-year-old woman) accepted acupuncture in the absence of medication because of her inability to tolerate the adverse events of anticonvulsant drugs. She achieved remission of pain 4 weeks later. The results of this study showed that acupuncture might be promising for controlling the RRNP in patients with cancer, especially who were intolerant or unresponsive to medications.

In this article, we investigate the existence of least energy sign-changing solutions for the following Schrödinger-Poisson system −Δu+V(x)u+K(x)ϕu=f(u),x∈R3,−Δϕ=K(x)u2,x∈R3,\left\{\begin{array}{ll}-\Delta u+V\left(x)u+K\left(x)\phi u=f\left(u),\hspace{1.0em}& x\in {{\mathbb{R}}}^{3},\\ -\Delta \phi =K\left(x){u}^{2},\hspace{1.0em}& 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.

Aiming at the problem of community discovery in relational networks, considering the strength of interaction between nodes and information seepage mechanism, an edge weight and connected component (EWCC) community discovery algorithm based on edge weight and connected branches is innovatively proposed.In order to verify effectiveness of the algorithm, firstly, five kinds of interactive two-layer network models are constructed.By analyzing influence of interaction degree of nodes between layers on the network topology, 30 data sets generated under five kinds of two-layer network models are determined.Secondly, the real data set is selected to compare with GN algorithm and KL algorithm in the evaluation criteria of modularity, algorithm complexity and community division number.Experimental results show that EWCC algorithm has high accuracy.Then, the numerical simulation shows that with the weakening of interaction relationship between layers, the module degree is inversely proportional to number of communities, and the community division effect is better when node relationship between layers is weaker.Finally, as an application of the algorithm, the “user-APP” two-layer network is constructed based on empirical data, and the community is divided.

In the present paper, a class of Schrödinger equations is investigated, which can be stated as −Δu+V(x)u=f(u), x∈ℝN.- \Delta u + V(x)u = f(u),\;\;\;\;x \in {{\rm{\mathbb R}}^N}. If the external potential V is radial and coercive, then we give the local Ambrosetti-Rabinowitz super-linear condition on the nonlinearity term f ∈ C(ℝ, ℝ) which assures the problem has not only infinitely many radial sign-changing solutions, but also infinitely many non-radial sign-changing solutions.

BACKGROUND As an important mineralization element, mercury is widely distributed in different geological bodies and participates in diagenesis and mineralization. With the rapid development of mass spectrometry technology, the field of mercury isotope geochemistry has made remarkable progress. Mercury isotopes have been widely used to trace the biogeochemical processes of the earth's surface and mercury pollution. In recent years, mercury isotopes have been applied to reveal the evolution of planets, identify large igneous provinces in geological history, and trace the sources of mineral deposits. OBJECTIVES To summarize the mercury isotope compositions of different geological reservoirs (meteorites, terrestrial rocks, coal, sediments, volcanic emissions, epithermal deposits) and investigate the factors controlling the Hg isotope fractionation during ore-forming processes in epithermal deposits. METHODS Literature reviewed that included published data from this research group and others. RESULTS Based on previous studies, the isotope composition of mercury in different geological reservoirs was systematically studied. The mercury isotopic composition of geological reservoirs such as meteorites, magmatic rocks, metamorphic rocks, sedimentary rocks, and volcanic gases varied greatly, and some samples also contained non-mass fractionation information. The occurrence and isotopic composition characteristics of low-temperature hydrothermal deposits (modern hot springs, mercury deposits, lead-zinc deposits, antimony deposits, gold deposits) was the focus of this review, and the basic framework of the mercury isotope system construction. Combined with the latest research results, a comprehensive summary of the mercury isotope fractionation mechanism that may have occurred in the mineralization process of the deposit was carried out. The mass fractionation of mercury isotopes in hydrothermal deposits may be caused by fluid volatilization or boiling, condensation, redox reactions, and sulfide precipitation. The non-mass fractionation of mercury isotopes in rocks and ores may be the product of mercury photochemistry during the geological history, or the inheritance of a specific source rock information. CONCLUSIONS In the future, mercury isotope has great application potential in tracing the ore-forming source of low-temperature hydrothermal deposits and characterizing the evolution of ore-forming fluids.

In this paper, we investigate the non-autonomous Choquard equation

Ubiquitin is a conserved signaling protein. It is able to interact with thousands of target proteins specifically and perform its biological functions via non-covalent interactions. Ubiquilin-1 (Ubql-1) and Rad23A are two transport factors in the protein-degradation process, and both contain ubiquitin-associated domains (UBAs). S65 of the ubiquitin can be phosphorylated specifically by the kinase PINK1, and the phosphorylated ubiquitin has two interchangeable conformations in solution, the relaxed conformation and the retracted conformation. In this work, nuclear magnetic resonance (NMR) methods were used to characterize the regulation of interactions between the ubiquitin and the UBAs of Rad23A/Ubql-1 by phosphorylation. The results indicated that the UBAs selectively interacted with the relaxed conformation of the phosphorylated ubiquitin without affecting the affinity. When interacting with the UBA of Ubql-1, inter-protein binding appeared to be able to promote the transferring from the retracted conformation to the relaxed conformation.

In this paper, we investigate a class of Kirchhoff type problems in ℝ3{\mathbb{R}^{3}} involving a critical nonlinearity, namely,

In this article, we establish the existence of bound state solutions for a class of quasilinear Schrödinger equations whose nonlinear term is asymptotically linear in ℝN{\mathbb{R}^{N}}. After changing the variables, the quasilinear equation becomes a semilinear equation, whose respective associated functional is well defined in H1⁢(ℝN){H^{1}(\mathbb{R}^{N})}. The proofs are based on the Pohozaev manifold and a linking theorem.

In addition to a face-to-face classroom learning environment, today's learners in higher education are likely to have also experienced a blended learning or an online learning environment. These learning environments not only differ in their delivery modes, but also learning activities, class interactions, assessment approaches, etc. Learners tend to have differing perceptions about the effectiveness of different learning environments. This study therefore investigates whether the reasons learners like or dislike a learning environment reveal learner characteristics that may explain why some learners are more inclined towards a particular learning environment. This study also examines whether learner demographics influence learner characteristics and their preference for a particular learning environment. Using an exploratory sequential mixed methods research design, this study first conducted several focus group discussions and then administered an online questionnaire survey to collect input from students at a local university. Analyses derived four learner characteristics (i.e. desire for direct support, digital readiness, learning independence, and online hesitancy) based on the reasons why the students liked or disliked face-to-face classroom learning, blended learning, or online learning environments. A cluster analysis further distinguished the students into three groups (i.e. classroom learners, insecure learners, and online learners) based on the four learner characteristics. Analyses also found that learners' demographics largely had no effect on learners' characteristics and their preference for a particular learning environment. The findings suggest that learner characteristics may provide a clue to why certain learners have a preference for a face-to-face classroom learning, a blended learning, or an online learning environment. A better understanding of the relationship between learner characteristics and learners' inclination towards a particular learning environment can be helpful to educational institutions and academics to design a range of engaging learning activities for learners with different characteristics.

Some existence theorems are obtained for infinitely many periodic solutions of ordinary p-Laplacian systems by minimax methods in critical point theory.