This article concerns the following Klein–Gordon–Maxwell system −Δu+u−(2ω+ϕ)ϕu=f(x,u),inR3,Δϕ=(ω+ϕ)u2,inR3,where ω>0 is a constant. When f satisfies a weaker 1-superlinear condition, existence ...results for nontrivial solutions and a sequence of high energy solutions are obtained by the Mountain Pass Theorem and Symmetric Mountain Pass Theorem. Our result completes some recent works concerning research on solutions of this system.
In this work, we study the existence of nontrivial solutions for the following class of partial differential inclusion problem: -Δu+1+λV(x)u∈∂tF(x,u)inRN,(Pλ)where N≥1, λ>0, V is a continuous ...function verifying some conditions and ∂tF(x,u) is the generalized gradient of F(x, t) with respect to t. Assuming that F(x, t) is a mensurable function for each t∈R and locally Lipschitzian for each x∈RN, we have applied variational methods for locally Lipschitz functionals to get a solution for (Pλ) when λ is large enough.
In this paper, we study the existence of ground state solutions for the nonlinear fractional Schrödinger–Poisson system with critical Sobolev ...exponent{(−Δ)su+V(x)u+ϕu=μ|u|q−1u+|u|2s⁎−2u,in R3,(−Δ)tϕ=u2,in R3, where μ∈R+ is a parameter, 1<q<2s⁎−1=3+2s3−2s, s,t∈(0,1) and 2s+2t>3. Under certain assumptions on V(x), using the method of Pohozaev–Nehari manifold and the arguments of Brezis–Nirenberg, the monotonic trick and global compactness Lemma, we prove the existence of a nontrivial ground state solution.
In this paper, we study the following nonlocal problem in RN−a−ϵ∫RN|Δu|2dxΔu+(1+λf(x))u=|u|p−2u,where a>0, 2<p<2∗=2N∕(N−2) with N≥3, ϵ>0 is small enough, and the parameter λ>0. Under some assumptions ...on f(x), we prove the existence of ground state solutions for the problem when λ is large enough via variational methods. In addition, we obtain some concentration behaviors of these solutions as λ→+∞.
Matrix-product states have become the de facto standard for the representation of one-dimensional quantum many body states. During the last few years, numerous new methods have been introduced to ...evaluate the time evolution of a matrix-product state. Here, we will review and summarize the recent work on this topic as applied to finite quantum systems. We will explain and compare the different methods available to construct a time-evolved matrix-product state, namely the time-evolving block decimation, the MPO WI,II method, the global Krylov method, the local Krylov method and the one- and two-site time-dependent variational principle. We will also apply these methods to four different representative examples of current problem settings in condensed matter physics.
•Introductory review of time-evolution methods for matrix-product states.•Detailed description of five state-of-the-art algorithms.•Extended comparison of numerical examples.•Extended error analysis of each algorithm.•Pseudo-code examples towards an implementation where necessary.
In this paper, we study the autonomous Choquard equation−Δu+u=(Iα⁎F(u))f(u),inRN, where N≥3, 0<α<N, Iα is a Riesz potential, and f∈C(R,R) satisfies the general Berestycki–Lions conditions. In Sec. 2, ...combining constrained variational method with deformation lemma, we obtain a ground state solution of Pohoz̆aev type for the above equation. In Sec. 3, using non-Nehari manifold method, we prove that the above equation has a ground state solution of Nehari type. The results improve some ones in 12.
In this paper, we address the image denoising problem in presence of speckle degradation typically arising in ultra-sound images. Variational methods and Convolutional Neural Networks (CNNs) are ...considered well-established methods for specific noise types, such as Gaussian and Poisson noise. Nonetheless, the advances achieved by these two classes of strategies are limited when tackling the de-speckle problem. In fact, variational methods for speckle removal typically amounts to solve a non-convex functional with the related issues from the convergence viewpoint; on the other hand, the lack of large datasets of noise-free ultra-sound images has not allowed the extension of the state-of-the-art CNN denoiser methods to the case of speckle degradation. Here, we aim at combining the classical variational methods with the predictive properties of CNNs by considering a weighted total variation regularized model; the local weights are obtained as the output of a statistically inspired neural network that is trained on a small and composite dataset of natural and synthetic images. The resulting non-convex variational model, which is minimized by means of the Alternating Direction Method of Multipliers (ADMM) is proven to converge to a stationary point. Numerical tests show the effectiveness of our approach for the denoising of natural and satellite images.
•An hybrid strategy for the despeckling problem combining convolutional neural networks and variational methods is proposed.•The proposed neural architecture is statistically-inspired and not requires the training on a large data-set of images.•A proof of convergence to a stationary point for the alternating direction method is given.
The Stochastic Variational Method (SVM) is used to show that the effective mass model correctly estimates the binding energies of excitons and trions but fails to predict the experimental binding ...energy of the biexciton. Using high-accuracy variational calculations, it is demonstrated that the biexciton binding energy in transition metal dichalcogenides is smaller than the trion binding energy, contradicting experimental findings. It is also shown that the biexciton has bound excited states and that the binding energy of the L = 0 excited state is in very good agreement with experimental data. This excited state corresponds to a hole attached to a negative trion and may be a possible resolution of the discrepancy between theory and experiment.
By means of variational methods and systematic numerical analysis, we demonstrate the existence of metastable solitons in three dimensional (3D) free space, in the context of binary atomic ...condensates combining contact self-attraction and spin-orbit coupling, which can be engineered by available experimental techniques. Depending on the relative strength of the intra- and intercomponent attraction, the stable solitons feature a semivortex or mixed-mode structure. In spite of the fact that the local cubic self-attraction gives rise to the supercritical collapse in 3D, and hence the setting produces no true ground state, the solitons are stable against small perturbations, motion, and collisions.
The existence of a positive solution to a Kirchhoff type problem on RN is proved by using variational methods, and the new result does not require usual compactness conditions. A cut-off functional ...is utilized to obtain the bounded Palais–Smale sequences.
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