For a class of semi-linear elliptic systems of Gross-Pitaevskii type with Dirichlet boundary conditions, {Delta ui,(beta)=f(i,beta)(x, u(i,beta))-beta ui,beta(k)& sum;(j/=i)a(ij)u(j,beta)(2),u(i,beta)>0 in Omega, ui,beta=0 on partial derivative Omega where N >= 1,1 <= i <= k with k >= 2,beta>0 and Omega is a domain in with partial derivative Omega bounded or satisfying a uniform exterior sphere condition, we prove that uniform boundedness of the solutions implies their uniform global Lipschitz boundedness as beta ->+infinity . As a consequence, the limiting profile is globally Lipschitz continuous in Omega as beta ->+infinity . This improves interesting uniform H & ouml;lder regularity results by Terracini et al. (Comm. Pure Appl. Math., 2010) to the optimal case and cover the one dimension problem, here the space dimension can be arbitrary and the domain can be unbounded. We also obtain the uniform interior Lipschitz bounds of the solutions, and develop the result of Soave and Zilio (Arch. Ration. Mech. Anal., 2015) in a general setting (especially we remove the pre-compactness assumption on and the restriction of N, which are technical therein). The proofs are based on blow-up analysis and Alt-Caffarelli-Friedman type monotonicity formulae both in the domain and near the domain boundary. 2026 ElsevierInc. Allrights are reserved, including those for text and data mining, AItraining ,and similar technologies

In this paper, we analyze new dynamical phenomena of the solutions to the Cauchy problem of Schrödinger equations with Chern–Simons gauge field. This model arises in the non-relativistic gauge theory and captures a strong nonlocal effect due to the presence of the covariant derivative operators. We study the existence and classification of the initial data in H^1(ℝ^2) which generate either global or finite time blow-up solution to the problem under the Coulomb gauge condition, by developing a delicate variational characterization and some suitable scaling properties of the energy functional. In particular, we prove that there exist ϕ∈ H^1(ℝ^2) and real numbers τ _1 < τ _2 , such that the solution u with u(0,x) = τϕ (τ x) exists globally for any τ < τ _1 and blows up in finite time provided τ > τ _2 . The Chern–Simons gauge potentials make the problem interesting and different from the classical Schrödinger equations.

In this paper, we are concerned with the non-existence of positive solutions for higher order Hartree type system {[ (-Δ )^m u=(1/|x|^σ*v^p)v^p-1, x∈ℝ^N,; (-Δ )^m v=(1/|x|^σ*u^q)u^q-1, x∈ℝ^N, ]. where N>2m , m≥ 1 , 0<σ 1 . In the first step, we establish the equivalence between partial differential system and integral system by using super poly-harmonic properties. In addition, we prove that the above system has no positive sup-solution under a Serrin-type condition. By the method of moving sphere, we established the Liouville-type theorem and derive a classification of nonnegative solutions for the integral system in ℝ^N . As an application of Liouville-type theorem, through the Doubling Lemma, we obtain the singularity estimates of the nonnegative solutions on a bounded domain.

In this paper, we investigate the normalized solutions searching the conditions (with rho, N, p) for the existence of solutions to the following elliptic equation { -triangle U +lambda U = |x|(alpha)|U|Up-1, in ohm U = 0, on partial derivative ohm integral (ohm) U-2 dx = rho, where N >3, ohm C R-N is is a ball or an annulus, rho> 0, 2 < alpha < + infinity and 1 < p < p(alpha) := N+2+2 alpha /N-2 . More precisely, we classify the problem into three cases, i.e., p is L-2-sub critical (p < 1 + 4+2 alpha/N), L-2-supercritical (p = 1 + 4+2 alpha /N ) based on a general Gagliardo-Nirenberg inequality and an adapted pointwise blow-up analysis. When pis L-2-subcritical, the problem admits solutions for every rho >0. In the L-2-critical and supercritical case, we show that for any k is an element of N the problem admits solutions with Morse index bounded above bykonlyif rho is sufficiently small. Furthermore, we present existence results for certain ranges of rho, which can be estimated in terms of the Dirichlet eigenvalues of-triangle inH(0)(1),(rad)(ohm).

We study the existence of subharmonic and time-periodic solutions with prescribed minimal period for wave equations on a disk with radius R in ℝ^2 . Such issues were posed by Rabinowitz as open problems in the context of finite dimensional Hamiltonian systems. The wave equation has an infinite dimensional Hamiltonian and its energy functional is strongly indefinite, which create considerable challenges. Our contributions are threefold: (i) By a deep analysis of Bessel functions, we obtain new spectral properties and a compact embedding theorem to the wave operator on a disk. (ii) For any m ∈ℤ_+ and T_m = R/m , we study the existence of T_m -periodic solutions for the focusing problem and show the nT_1 -periodic solutions (called subharmonics) are distinct for different n ∈ℤ_+ . (iii) Under some suitable restrictions, we prove that the solutions for the defocusing problem have T_m as the minimal period. This is the first result about the minimal period for the solutions and subharmonics of wave equations in multidimensional space.

We consider the existence of normalized ground states to the following Sobolev critical Schrodinger systems with linear and nonlinear couplings. { -triangle v(1) + mu (1)v(1) - kappa v(2 )= |v(1)|(2 & lowast;-2)v(1) +xi alpha|v(1)|(alpha-2)|v(2)|beta(v1), in R-N, -triangle v(2) + mu 2v(2) - kappa v(1) = |v(2)|(2 & lowast;-2)v(2) +xi beta|v(1)|alpha|v(2)|beta-2(v2), in R-N, integral(RN) v(1)(2)dx= a(2)(1),integral(RN) v(2)(2)dx = a(2)(2), where N = 3 or 4, a1, a2 > 0, kappa is an element of R, xi>0, alpha>1, beta>1, alpha+beta is an element of (2, 2 & lowast;]\{2+ 4/N } and 2 & lowast; = 2N /N-2. Firstly, in the case alpha+ beta<2 + 4/N, by Ekeland's variational principle, we prove for any xi > 0, kappa > 0, the system admits a radially symmetric normalized solution provided a1, a2 are small; secondly, when 2 + 4N 0, there are two positive constants xi(& lowast;& lowast;) and xi(& lowast;) with xi(& lowast;& lowast; )> xi(& lowast;) > 0 such that for any xi is an element of (xi(& lowast;), xi(& lowast;& lowast;)), the system possesses a radially symmetric normalized ground state solution whenever kappa is an element of (0, kappa) for some kappa > 0. In both cases above, we overcome the lack of compactness of the L2-norm by virtue of some delicate energy estimates. Finally, when alpha + beta = 2(& lowast;), we prove that the system does not have any nontrivial nonnegative solutions.

We investigate the existence and blow-up profile of normalized solutions to the fractional nonlinear Schrodinger equation { (-Delta)(s)u + V (x)u + lambda u =|u| 4s/N u; in R-N, integral R-N |u|(2) dx = alpha, with N >= 2; s is an element of (0; 1); alpha > 0, lambda is an element of R, and negative potentials V (x). Firstly, we prove the existence and nonexistence of normalized solutions for negative potentials V is an element of L-p(R-N) + L-q (R-N) with N/2s <= p < q < infinity in the mass-critical setting. Secondly, we refine the nonexistence results and analyze the asymptotic behavior of the minimizers {u alpha} as alpha NE arrow alpha* for two types of potentials: a bounded potential V (x) = eta/(1+|x|(2))(s), and a singular potential V (x) = -gamma(1)|x - a(1)|(-m1) -gamma(2) |x - a(2)|(-m2). Our study provides precise energy estimates and sharp blow-up rates in the convergence of minimizers {u(alpha)}.

In this paper, we investigate the monotonicity of solutions to the nonlocal Monge-Ampère system {[ D^θ _s u(x)=f(u(x),v(x)), x ∈Ω , D^θ _t v(x)=g(u(x),v(x)), x ∈Ω , ]. where 00 , n≥ 2 , Ω⊆ℝ^n is a bounded domain which is convex in x_n -direction or the whole space, D^θ _α is the nonlocal Monge-Ampère operator ( α =s,t ), and f, g∈ C^1(ℝ^2) . We use the sliding method to prove any solution (u, v) of the system is strictly increasing in Ω with respect to x_n under some suitable conditions on f and g. The proof involves the idea that estimates the singular integrals along a sequence of approximate maximum points.

In this paper, we study nonlinearly coupled Schrodinger systems {-Delta u + lambda(1)u = mu(1)u(3) + beta(x)uv(2) -Delta v + lambda(2)v = mu(2)v(3) + beta(x)u(2)v in R-3 satisfying the normalization integral(R3) |u|(2) = a(1)(2) and integral(3)(R) |v|(2) = a(2)(2), where a(1), a(2), mu(1), mu(2) > 0 and beta(x) is an element of L infinity(R-3). We use concentration compactness principle to prove the existence of normalized solutions and give a criterion of the existence of normalized ground state solutions. We also construct a limit system to get the asymptotic behavior of the normalized ground states as beta(x) equivalent to beta is large enough. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Deep ultraviolet coherent light, particularly at the wavelength of 193 nm, has become indispensable for semiconductor lithography. We present a compact solid-state nanosecond pulsed laser system capable of generating 193-nm coherent light at the repetition rate of 6 kHz. One part of the 1030-nm laser from the home-made Yb:YAG crystal amplifier is divided to generate 258 nm laser (1.2 W) by fourth-harmonic generation, and the rest is used to pump an optical parametric amplifier producing 1553 nm laser (700 mW). Frequency mixing of these beams in cascaded LiB3O5 crystals yields a 193-nm laser with 70-mW average power and a linewidth of less than 880 MHz. By introducing a spiral phase plate to the 1553-nm beam before frequency mixing, we generate a vortex beam carrying orbital angular momentum. This is, to our knowledge, the first demonstration of a 193-nm vortex beam generated from a solid-state laser. Such a beam could be valuable for seeding hybrid ArF excimer lasers and has potential applications in wafer processing and defect inspection.

Consider the Kirchhoff equation with Hartree type nonlinearity - ( a + b∫_ℝ^3| ∇ u |^2)Δ u - λ u = μ| u |^q - 2u + ( I_α * | u |^3 + α )| u |^1 + αu in ℝ^3, where a, b > 0, λ, μ ∈ ℝ, 2 < q < 6, 0 < α < 3, and Iα is the Riesz potential integral operator of order α. Solutions with prescribed mass u_L^2(ℝ^3) = c > 0 , also known as normalized solutions, are of particular interest in the current paper. Under various assumptions on μ, c and q, we establish the existence, nonexistence and asymptotic behavior of normalized solutions for the above elliptic equation.

In this paper, we investigate the dynamics of geophysically stratified water waves with constant vorticity near the equator. Firstly, we prove that the bounded solutions of the three-dimensional inviscid gravity water wave governed by the f-plane approximation equation are two-dimensional. Moreover, we establish the two-dimensional characteristics of bounded solutions for gravity-capillary water waves and gravity water waves with rigid lids. Secondly, we focus on rigid lid case of gravity and gravity-capillary stratified water waves controlled by the β -plane approximation. Here, we find that the only flow with a constant vorticity vector is a stationary flow with a flat interface and a vanishing velocity field.

This paper is devoted to the study of normalized solutions to p-Laplacian equation involving Sobolev critical exponent. Under the focussing condition, we obtain existence of ground states in mass subcritical case, mass critical case and mass supercritical case, respectively. Furthermore, we show some asymptotic behaviors of ground states. Meanwhile, under the defocussing condition, we prove some non-existence results.

We classify the solutions of the planar weighted Allen-Cahn equation arising from the critical equivariant SO(4) Yang-Mills problem. We first give a complete classification for radially symmetric solutions, and then without symmetry assumptions, we show bounded classical solutions are radially symmetric and also classify them.

In this paper, we are concerned with the following elliptic equation involving a general gradient nonlinearity:−Δv=μ|v|p−1v+G(∇v)inΩ, where n≥1, either Ω=Rn is the whole space or Ω=R+n={x=(x1,…,xn)∈Rn:xn>0} is the half space, μ≥0, p∈R and G:Rn→R satisfies some suitable conditions. One particular case of G is G(γ)=∑i=1kai|γ|pi with ai>0 and pi>1 for any 1≤i≤k. When Ω=Rn, we establish a Bernstein estimate to prove some Liouville-type results of the equation under certain conditions on μ and p. When Ω=R+n, by combining the Bernstein estimates with a moving planes argument, we obtain the one-dimensional symmetry of the solution v with v=0 on ∂R+n. We note that the exponent p is allowed to be negative in our results. Finally, we obtain a gradient estimate in some special cases with μ>0 and p>1.

In this paper, we study Liouville-type theorems for fractional Hardy–Hénon elliptic systems with weights. Because the weights are singular at zero, we firstly prove that classical solutions for systems in ℝ^N \{0} are also distributional solutions in ℝ^N . Then we study the equivalence between the fractional Hardy–Hénon system and a proper integral system, and we obtain new Liouville-type theorems for supersolutions and solutions by the method of integral estimates and scaling spheres respectively.

In this paper, we are concerned with the monotonic and symmetric properties of convex solutions Monge-Amp\`ere systems for instance, considering \begin{equation*} \det(D^2u^i)=f^i(x,{\bf u},\nabla u^i), \ 1\leq i\leq m, \end{equation*} over unbounded domains of various cases, including the whole spaces $\mathbb{R}^n$, the half spaces $\mathbb{R}^n_+$ and the unbounded tube shape domains in $\mathbb{R}^n$. We obtain monotonic and symmetric properties of the solutions to the problem with respect to the geometry of domains and the monotonic and symmetric properties of right-hand side terms. The proof is based on carefully using the moving plane method together with various maximum principles and Hopf's lemmas.

In this paper we study the nonlinear Bose–Einstein condensates Schrödinger system -Δ u_1-λ _1 u_1=μ _1 u_1^3+β u_1u_2^2+ κ (x) u_2 in ℝ^3, -Δ u_2-λ _2 u_2=μ _2 u_2^3+β u_1^2u_2+ κ (x) u_1 in ℝ^3, ∫ _ℝ^3 u_1^2=a_1^2, ∫ _ℝ^3 u_2^2=a_2^2, where a_1 , a_2 , μ _1 , μ _2 , κ =κ (x)>0 , β <0 , and λ _1 , λ _2 are Lagrangian multipliers. We use the Ekeland variational principle and the minimax method on manifold to prove that this system has a solution that is radially symmetric and positive.

This paper considers the spreading speed of cooperative nonlocal dispersal system with irreducible reaction functions and non-uniform initial data. Here the non-uniformity means that all components of initial data decay exponentially but their decay rates are different. It is well-known that in a monostable reaction-diffusion or nonlocal dispersal equation, different decay rates of initial data yield different spreading speeds. In this paper, we show that due to the cooperation and irreducibility of reaction functions, all components of the solution with non-uniform initial data will possess a uniform spreading speed which non-increasingly depends only on the smallest decay rate of initial data. The nonincreasing property of the uniform spreading speed further implies that the component with the smallest decay rate can accelerate the spatial propagation of other components. In addition, all the methods in this paper can be carried over to the cooperative system with classical diffusion (i.e. random diffusion).

A 60-mW solid-state deep ultraviolet (DUV) laser at 193 nm with narrow linewidth is obtained with two stages of sum frequency generation in LBO crystals. The pump lasers, at 258 and 1553 nm, are derived from a homemade Yb-hybrid laser employing fourth-harmonic generation and Er-doped fiber laser, respectively. The Yb-hybrid laser, finally, is power scaling by a 2 mm x 2 mm x 30 mm Yb:YAG bulk crystal. Accompanied by the generated 220-mW DUV laser at 221 nm, the 193-nm laser delivers an average power of 60 mW with a pulse duration of 4.6 ns, a repetition rate of 6 kHz, and a linewidth of similar to 640MHz. To the best of our knowledge, this is the highest power of 193- and 221-nm laser generated by an LBO crystal ever reported as well as the narrowest linewidth of 193-nm laser by it. Remarkably, the conversion efficiency reaches 27% for 221 to 193 nm and 3% for 258 to 193 nm, which are the highest efficiency values reported to date. We demonstrate the huge potential of LBO crystals for producing hundreds of milliwatt or even watt level 193-nm laser, which also paves a brand-new way to generate other DUV laser wavelengths.