This paper considers the existence of multiple normalized solutions of the following Schrödinger-Choquard equation - Δ u = λ u + k(ε x) (I_α * |u|^q) |u|^q-2u + μ (I_α * |u|^p) |u|^p-2u, x ∈ℝ^N, ∫_ℝ^N |u|^2 dx = c^2, x ∈ℝ^N. where c, ε, μ > 0, N ≥ 3, α, ∈ (0; N), N + α N < q < 1 + α + 2 N < p ≤N + αN - 2 , λ ∈ ℝ is a Lagrange multiplier which is unknown, Iα is the Riesz potential, k:ℝN → [0; ∞) is a continuous and positive function. When ε is small enough, we prove that the numbers of normalized solutions are at least the numbers of global maximum points of k by Ekeland’s variational principle and truncated skill.

PurposeThis study aims to investigate the impact of price protection policies on sellers' pricing and profits on e-commerce platforms, and to analyze the effectiveness of these policies under different scenarios. The goal is to provide sellers with a basis for their pricing strategies and assist them in deciding whether to implement price protection policies.Design/methodology/approachThis paper constructs a two-period choice model by using the concept of rational expectations equilibrium, and analyzes factors such as the purchase decisions of strategic consumers, demand and seller profits. The research method includes a profit comparison under four scenarios: the benchmark scenario (B), considering only network externalities (N), considering only price protection policies (P) and considering both network externalities and price protection policies (NP).FindingsBy comparing profits across different scenarios, the study finds that price protection policies positively affect sellers' pricing and profits under specific conditions. The study identifies the conditions under which price protection policies are effective.Research limitations/implicationsThe limitations of this study lie in considering only four scenarios. Future research could expand to include more variables, such as other consumer behavior patterns.Practical implicationsThe findings offer valuable insights for e-commerce platform sellers in formulating pricing strategies and implementing price protection policies.Originality/valueThe originality of this paper lies in considering strategic consumers and network externalities. It fills a research gap in the field of e-commerce pricing.

In this paper, we mainly construct local solution curves for the two-dimensional steady compactly supported incompressible Euler equations with free boundaries and constant vorticity. Our work is distinguished from most existing studies on two-dimensional steady water waves by its focus on perturbations near annular flows, rather than laminar flows. More precisely, we consider three classes of steady Euler flows with compact support, corresponding to partially overdetermined, two-phase overdetermined, and overdetermined elliptic problems. The primary contribution of our work is threefold. For each class, we first establish the flexibility result (i.e., the existence of nontrivial admissible domains) via shape derivatives and local bifurcation theory. Second, we give and discuss the corresponding rigidity result respectively. Third, we apply the implicit function theorem to demonstrate the stability of standard annular flows under perturbations of the Neumann boundary condition. Our results also offer novel insights into the theory of elliptic overdetermined problems.

The purpose of this paper is to investigate the existence of normalized solutions for the coupled elliptic system with quadratic nonlinearity. In the first scenario, under an explicit smallness assumption on the potential, we establish the existence of a mountain pass solution at a positive energy level. In the second scenario, if the mass is smaller and depends on the potential, we identify a local minimizer at a negative energy level and a mountain pass solution at a positive energy level.

This paper investigates the global behavior of classical solutions to a parabolic Lane-Emden system. Under subcritical conditions, we first establish an intersection structure between regular and singular solutions of the associated elliptic system. Based on this, we construct global upper solutions and thereby prove the uniform decay of parabolic solutions at infinity. Moreover, in the degenerate case q = 1, we introduce a Sturm-type argument and, for the first time, establish the identity between weak continuous upper solutions and regular solutions of the elliptic system, which further extends the above decay result. Second, we derive the precise decay rates of solutions under subcritical and Joseph-Lundgren conditions through a case-by-case analysis. Finally, we construct families of initial data that lead to finite-time blow-up under both subcritical and Joseph-Lundgren conditions.

In this paper, we study the global existence and asymptotic behavior of solutions to the energy-critical Schr & ouml;dinger system with quadratic nonlinearity. First, we establish global well-posedness and characterize the long-time dynamics of solutions when the energy lies below a critical threshold. Second, we prove the existence of finite-time blow-up solutions for initial data with negative energy. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

In this paper we study the Cauchy problem for semilinear parabolic system with nonconstant coefficient singular initial data {[ U_t-Δ U=μ _1|U|^p-1U+β |U|^r-1U|V|^r+1, x∈ℝ^N,t>0,; V_t-Δ V=μ _2|V|^p-1V+β |U|^r+1|V|^r-1V, x∈ℝ^N,t>0,; U(x,0)=λ _1 a(x/|x|)|x|^-2/(p-1), x∈ℝ^N∖{0},; V(x,0)=λ _2 b(x/|x|)|x|^-2/(p-1), x∈ℝ^N∖{0},; ]. where N≥ 2 , p=2r+1 , μ _1,μ _2,β >0 , λ _1,λ _2>0 are constant parameters, a≥ 0≢0 , b≥ 0≢0 . We demonstrate that when 2

This study focuses on understanding wave propagation in dispersive and nonlinear media by analyzing the dynamic behavior of the modified KdV–Zakharov–Kuznetsov (mKdV–ZK) equation. Building on the extensively researched KdV-ZK equation, we examine solitary wave solutions of the (3+1)-dimensional mKdV–ZK equation using the extended modified Sardar sub-equation method. As a result, a diverse array of novel soliton solutions are obtained, including multi-peak solitons, kink and anti-kink waves, dark and bright solitons, and breather waves. These analytical solutions are derived in terms of key physical quantities such as electrostatic field potential, quantum statistical pressure, electric fields, and magnetic fields. Graphical representations are provided to illustrate the distinct dynamical features of the obtained wave solutions, highlighting their relevance to plasma physics, nonlinear optics, and electromagnetic wave propagation. The results confirm that the extended modified Sardar sub-equation method is an efficient and versatile tool for solving higher-dimensional nonlinear partial differential equations, with broad applicability to nonlinear wave systems in applied mathematics and physical sciences.

Mathematical modeling using non-integer order derivatives is gaining attention among scientists and researchers. Traditional models only use integer order derivatives, which limits their ability to accurately model real-life problems. New concepts of fractal-fractional derivatives, such as the exponential decay and Mittag-Leffler kernel, have been introduced to overcome the limitations of previous fractional order derivatives. In this study, we utilize fractal-fractional derivative, which is a new class of fractional derivative with power Law kernel to generalize two-dimensional degenerate parabolic equation arising in the spatial diffusion of biological population to fractal dimension and provide semi-analytical solutions using Laplace transforms. We discuss various cases of population models and provide graphical representations of the solutions. Furthermore, we develop a relationship between fractal-fractional derivative and the Caputo fractional derivative, and provide the results in terms of a lemma.

This paper is devoted to the existence of solution for logarithmic Schr & ouml;dinger equations. In contrast to most studies, we consider that the potential is indefinite. With the help of Morse theory, the existence of nontrivial solutions for the above problem is obtained.

The objective of our paper is to investigate fractional elliptic equations of the form (-Δ)^s u=λ/(a-u)^2 within a bounded domain Ω, subject to zero Dirichlet boundary conditions. Here, s∈(0,1), λ>0, and the function a vanishes at the boundary while satisfying additional conditions. This problem originates from Micro-Electromechanical Systems (MEMS) devices, particularly when the elastic membrane makes contact with the ground plate at the boundary. We establish both existence and nonexistence results, illustrating how the boundary decay of the membrane influences the solutions and pull-in voltage.

Geometric confinement is known to modify single-particle dynamics through effective potentials, yet its imprint on the interacting quantum vacuum remains largely unexplored. In this work, we investigate the Maxwell-Klein-Gordon system constrained to curved surfaces and demonstrate that the geometric potential Sigma geom(r) acts as a local renormalization environment. Going beyond standard approximations, we derive an exact analytical framework where extrinsic curvature modifies the scalar loop spectrum, entering the vacuum polarization as a position-dependent mass correction M2(r) -> m2 + Sigma geom(r). This induces a finite, gauge-invariant 'geometry-induced running' of the electromagnetic response. In the long-wavelength regime (divided by Q divided by R << 1), we obtain a closed-form expression for the relative frequency shift Delta omega/omega, governed by the overlap between the electric energy density and the geometric potential. Applying this formalism to Gaussian bumps, cylindrical shells, and tori, we identify distinct spectral signatures that distinguish these quantum loop corrections from classical geometric optics. Our results suggest that spatial curvature can serve as a tunable knob for 'vacuum engineering,' offering measurable shifts in high-Q cavities and plasmonic systems relevant to current nanophotonic experiments.

This study provides a thorough investigation of the commensurate fractional-order Rucklidge system, addressing key limitations in existing research. We validate the chaotic behavior of the system by calculating the Lyapunov exponents and characterizing the fractal nature of the attractor using the Kaplan–Yorke dimension. A bifurcation diagram is obtained to examine the transitions between periodic and chaotic regimes, while the stability of the system is assessed using the fractional Routh–Hurwitz criterion. Additionally, we evaluate the offset boosting capability of the system, which allows for controlled manipulation of attractor positions without altering the system’s inherent dynamics. Furthermore, we quantify signal complexity using spectral entropy and C_0 complexity metrics. To assess the randomness and cryptographic potential of the generated sequences, we perform validation with the NIST SP 800-22 statistical test suite, confirming their suitability for secure applications. By utilizing the high entropy and sensitivity to parameter variations inherent to the fractional order Rucklidge system, we propose a novel image encryption scheme that integrates chaotic keystreams into the ChaCha20 algorithm. Experimental results demonstrate significant improvements in security, highlighting the system as a powerful tool for both nonlinear system analysis and cryptographic applications.

n this paper, we consider the fractional heat equation with critical exponent in R-n for n>6s, with s is an element of (0, 1), u(t) = (- Delta) (s) u + |u|4s/n-2s u , (x,t )is an element of R-n x R We construct a bubble-tower-type solution for both the forward and the backward problem by establishing the existence of a sign-changing solution with multiple blow-up at a single point, having the form u(x, t) = (1+0 (1)) (k) Sigma (j=i) (-1) (j-1) u j (t) (n-2s/2) u(x/u (j)(t)) as t -> +infinity and of a positive solution with multiple blow-up at a single point, having the form u(x, t) = (1+0 (1)) (k) Sigma (j=i) u j (t) (n-2s/2) u(x/u (j)(t)) as t -> +infinity respectively. Herek >= 2is a positive integer, u(y) =alpha (n, s) (1/1+|y|(2)) (n-2s/2) The research of L. Cai is partially supported by the National Natural Science Foundation of China No. 12501150, the Natural Science Foundation of the Jiangsu Province No. BK20250885, and the Natural Science Foundation of the Jiangsu Higher Institutions of China No. 25KJB110020.The research of J. Wang is partially supported by the National Key R&D Program of China2022YFA1005601 and the National Natural Science Foundation of China 12371114. There search of J.-C. Wei is partially supported by the National Key R&D Program of China2022YFA1005602, and the Hong Kong General Research Funds "New frontiers in singularity formations of nonlinear partial differential equations" and "On Fujita equation in critical and supercritical regime". The research of W. Yang is partially supported by the National Key R&D Program of China 2022YFA1006800, the NSFC Nos. 12171456, 12271369 and 12531010, the FDCT No. 0070/2024/RIA1, the Start-up Research Grant No. SRG2023-00067-FST, the Multi-Year Research Grant Nos. MYRG-GRG2024-00082-FST-UMDF and MYRG-GRG2025-00051-FST, and the UMDF No. TISF/2025/006/FST. where alpha (n, s) is a constant depending only on n and s, and u(j)t =beta j |t| (-alpha j) (1+0(1)) as t -> +infinity , alpha(j) = 1/ 2s (n-2s/ n-6s)j-1 - 1/2s , for certain positive numbers beta j , j=1 ,...k.

In this paper, we explore the existence of fully nontrivial solutions to the following nonlinear Brezis-Nirenberg Maxwell system {del x del xE(1)+lambda E-1(1)=kappa(1)|E-1|E-p-2(1)+beta|E-2|E-2(1), in Omega, del x del xE(2)+lambda E-2(2)=kappa(2)|E-2|E-p-2(2)+beta|E-1|E-2(2),in Omega, nu(1)xE(1)=0, nu(2)xE(2)=0, on partial derivative Omega, where lambda(i)<= 0,kappa(i)>0(i=1,2),beta>0, p is an element of(2,6], and Omega subset of R-3 is a simply connected, smooth, bounded Lipschitz domain with connected boundary, and nu(1),nu(2):partial derivative Omega -> R-3 are the exterior normal. This system originates from the time-harmonic Maxwell equations and possesses a variational structure. We address both general subcritical cases and Sobolev critical cases, establishing the existence of a fully nontrivial ground state solution with cylindrical symmetry. Additionally, we prove several properties of these solutions. To achieve this, we develop a new critical point theory that not only resolves the current problem but also facilitates the treatment of more general anisotropic media and other variational problems. Notably, our results provide a positive answer to the open problem posed by T. Bartsch and J. Mederski in [7, Page 982, Problem 3]. Moreover, from a purely mathematical perspective, we extend the partial results from N=3 to the case of N=4.

This paper is concerned with the existence of normalized solutions to the problem with combined nonlinearities {-Delta u + u/r(2) - lambda u - kappa vertical bar u vertical bar(q-2) u - vertical bar u vertical bar(p-2) u = 0, x = (y,z) is an element of R-K x RN-K, integral(RN)vertical bar u vertical bar(2)dx = alpha(2) > 0, where N >= 3, kappa is an element of R, r=vertical bar y vertical bar = (x(1)(2) + x(2)(2) + ... + x(K)(2))(1/2) and p,q is an element of(2,2*). Here lambda is a Lagrange multiplier, which appears due to the prescribed mass constraint vertical bar u vertical bar(2) = alpha. Under different assumptions about the parameter kappa and exponents of nonlinear terms p,q, we explore the existence of solutions to the above equation, including the identification of minimum solutions and the analysis of mountain path solutions. Moreover, we shall also study the solution of the related curl-curl equation {del x del x U - lambda U - kappa vertical bar U vertical bar Uq-2 - vertical bar U vertical bar Up-2=0, integral(RN)vertical bar U vertical bar(2)dx = alpha(2) > 0, which arises from the system of Maxwell equations and is crucial in nonlinear optics. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies

In this article, we are concerned with the existence, non-existence, and blow-up behavior of normalized ground state solutions for the mass critical Hartree-Fock type Schrödinger equation with rotation i ∂ t u = − Δ u + 2 V ( x ) u + 2 Ω L z u − λ u − b u ∫ R N ∣ u ( y ) ∣ 2 ∣ x − y ∣ 2 d y , ( t , x ) ∈ R × R N , u ( 0 , x ) = u 0 ( x ) , \left\{\begin{array}{l}i{\partial }_{t}u=-\Delta u+2V\left(x)u+2\Omega {L}_{z}u-\lambda u-bu\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\frac{{| u(y)| }^{2}}{{| x-y| }^{2}}{\rm{d}}y,\hspace{1em}\left(t,x)\in {\mathbb{R}}\times {{\mathbb{R}}}^{N},\hspace{1.0em}\\ u\left(0,x)={u}_{0}\left(x),\hspace{1.0em}\end{array}\right. where N ≥ 3 N\ge 3 , b > 0 b\gt 0 , V ( x ) = ∣ x ∣ 2 2 V\left(x)=\frac{{| x| }^{2}}{2} , L z {L}_{z} is the angular momentum operator with the critical rotational speed Ω = 1 \Omega =1 , and the constant λ \lambda is the unknown Lagrange multiplier. We prove that the L 2 {L}^{2} -constraint minimizers exist if and only if the parameter b b satisfies b < b * = ‖ U ‖ 2 2 b\lt {b}_{* }={\Vert U\Vert }_{2}^{2} , where U U is a positive radially symmetric ground state of − Δ u + u − u ∫ R N u 2 ( y ) ∣ x − y ∣ 2 d y = 0 -\Delta u+u-u{\int }_{{{\mathbb{R}}}^{N}}\frac{{u}^{2}(y)}{{| x-y| }^{2}}{\rm{d}}y=0 in R N {{\mathbb{R}}}^{N} . We also establish the orbital stability result of prescribed mass standing waves for the equation when b < b * b\lt {b}_{* } . When b b approaches b * {b}_{* } , the system collapses to a profile obtained from the optimizer of a Gagliardo-Nirenberg inequality.

In this paper, we establish the existence of Stokes waves with piecewise smooth vorticity in a two-dimensional, infinitely deep fluid domain. These waves represent traveling water waves propagating over sheared currents in a semi-infinite cylinder, where the vorticity may exhibit discontinuities. The analysis is carried out by applying a hodograph transformation, which reformulates the original free boundary problem into an abstract elliptic boundary value problem. Compared to previously studied steady water waves, the present setting introduces several novel features: the presence of an internal interface, an unbounded spatial domain, and a non-Fredholm linearized operator. To address these difficulties, we introduce a height function formulation, casting the problem as a transmission problem with suitable transmission conditions. A singular bifurcation approach is then employed, combining global bifurcation theory with Whyburns topological lemma. Along the global bifurcation branch, we show that the resulting wave profiles either attain arbitrarily large wave speed or approach horizontal stagnation.

We study the coupled Schrödinger equations with critical exponent on ℝ^3 ×𝕋. With the help of scaling argument and semivirial-vanishing technology, we obtain the existence and y-dependence of solution, the tori can be generalized to 1-dimensional compact Riemannian manifold. Moreover, the conclusion of this paper can be extended to systems with any number of components.

In this paper we investigate the global existence and asymptotical stability of solutions to a class of parabolic systems with homogeneous nonlinearity for both bounded and unbounded domains. First we prove both global existence and finite time blow-up of solutions of the system for different initial conditions by using the potential well method, and the asymptotic behavior of the solutions are also considered. On the other hand, we also obtain global existence and finite time blow-up of solutions for both Sobolev subcritical and critical cases. We use a method of comparing least energy levels with that of semitrivial solutions to overcome the difficulties here.