Narrow Results

In this paper, we investigate the existence of solutions to a $p$-Kirchhoff problem involving the $\psi$-Hilfer fractional derivative. Specifically, we reformulate the problem by analyzing the associated functional energy. We then construct three disjoint sets and demonstrate that the functional energy has a critical point in each of these sets. Finally, based on the definitions of these sets, we show that in the first set the solution is positive, in the second set the solution is negative, and in the third set, the solution changes sign.

In this paper, we consider the existence of periodic solutions of Lagrangian systems of the form $\frac{d}{dt}(\nabla \mathrm{\Phi }(\dot{u}(t)))+V_u(t,u(t))+\lambda G_u(t,u(t))=0$, where $\mathrm{\Phi }:{ℝ}^n\to [0,\infty )$ is a ${𝒢}$-function in the sense of Trudinger, $V,G:ℝ\times {ℝ}^n\to ℝ$ are $C^1$-smooth, $T$-periodic in the time variable $t$ and $\lambda$ is a real parameter. We prove the existence of a $T$-periodic solution $u_{\lambda }:ℝ\to {ℝ}^n$ for any sufficiently small $\left|\lambda \right|$, and show that the found solutions converge to a $T$-periodic solution of the unperturbed system if $\lambda$ tends to $0$. Let us stress that our theorem only makes a natural assumption on the potential $V$ and no additional assumption on $G$ except that it is continuously differentiable. Its proof requires to work in a rather unusual (mixed) Orlicz–Sobolev space setting, which bears several challenges.

This paper deals with the following fractional Choquard equation:

In this paper we establish a new mean field-type formulation to study the problem of prescribing Gaussian and geodesic curvatures on compact surfaces with boundary, which is equivalent to the following Liouville-type PDE with nonlinear Neumann conditions:

In this paper, we prove the existence, uniqueness and qualitative properties of heteroclinic solution for a class of autonomous quasilinear ordinary differential equations of the Allen–Cahn type given by

In this paper, we are interested in the existence and multiplicity of multi-bump solutions for critical Schrödinger equations with electromagnetic fields and logarithmic nonlinearity of the following type:

In this paper, we concern some qualitative properties of the following $(p,N)$-Laplacian equations with convolution term:

This paper is concerned with the existence of multiple normalized solutions for a class of Choquard equation involving the biharmonic operator and competing potentials in ${ℝ}^N$:

In nuclear physics, the relativistic mean-field theory describes the nucleus as a system of Dirac nucleons which interact via meson fields. In a static case and without nonlinear self-coupling of the σ meson, the relativistic mean-field equations become a system of Dirac equations where the potential is given by the meson and photon fields. The aim of this work is to prove the existence of solutions of these equations. We consider a minimization problem with constraints that involve negative spectral projectors and we apply the concentration-compactness lemma to find a minimizer of this problem. We show that this minimizer is a solution of the relativistic mean-field equations considered.

The multiconfiguration Dirac–Fock (MCDF) model uses a linear combination of Slater determinants to approximate the electronic N-body wave function of a relativistic molecular system, resulting in a coupled system of nonlinear eigenvalue equations, the MCDF equations. In this paper, we prove the existence of solutions of these equations in the weakly relativistic regime. First, using a new variational principle as well as the results of Lewin on the multiconfiguration non-relativistic model, and Esteban and Séré on the single-configuration relativistic model, we prove the existence of critical points for the associated energy functional, under the constraint that the occupation numbers are not too small. Then, this constraint can be removed in the weakly relativistic regime, and we obtain non-constrained critical points, i.e. solutions of the multiconfiguration Dirac–Fock equations.

This paper is concerned with the following Kirchhoff–Schrödinger–Poisson system:

A nonperturbative effective model is derived for the Higgs sector of the Standard Model which is described by a simple scalar theory. The renormalized couplings are determined by the derivatives of the Gaussian effective potential that are known to be the sum of infinite bubble graphs contributing to the vertex functions. A good agreement has been found with strong coupling lattice simulations when a comparison can be made.

The approach to the theory of many-particle interacting systems from a unified standpoint, based on the variational principle for free energy is reviewed. A systematic discussion is given of the approximate free energies of complex statistical systems. The analysis is centered around the variational principle of Bogoliubov for free energy in the context of its applications to various problems of statistical mechanics. The review presents a terse discussion of selected works carried out over the past few decades on the theory of many-particle interacting systems in terms of the variational inequalities. It is the purpose of this paper to discuss some of the general principles which form the mathematical background to this approach and to establish a connection of the variational technique with other methods, such as the method of the mean (or self-consistent) field in the many-body problem. The method is illustrated by applying it to various systems of many-particle interacting systems, such as Ising, Heisenberg and Hubbard models, superconducting (SC) and superfluid systems, etc. This work proposes a new, general and pedagogical presentation, intended both for those who are interested in basic aspects and for those who are interested in concrete applications.

We compute the energy spectrum of the ground state of a 2D Dirac electron in the presence of a Coulomb potential and a constant magnetic field perpendicular to the plane where the the electron is confined. With the help of a mixed-basis variational method we compute the wave function and the energy level and show how it depends on the magnetic field strength. We compare the results with those obtained numerically as well as in the non-relativistic limit.

We provide a tutorial on learning and inference in hidden Markov models in the context of the recent literature on Bayesian networks. This perspective makes it possible to consider novel generalizations of hidden Markov models with multiple hidden state variables, multiscale representations, and mixed discrete and continuous variables. Although exact inference in these generalizations is usually intractable, one can use approximate inference algorithms such as Markov chain sampling and variational methods. We describe how such methods are applied to these generalized hidden Markov models. We conclude this review with a discussion of Bayesian methods for model selection in generalized HMMs.

The existence of minimizers to a geometrically exact Cosserat planar shell model with microstructure is proven. The membrane energy is a quadratic, uniformly Legendre–Hadamard elliptic energy in contrast to traditional membrane energies. The bending contribution is augmented by a curvature term representing the interaction of the rotational microstructure in the Cosserat theory. The model includes non-classical size effects, transverse shear resistance, drilling degrees of freedom and accounts implicitly for thickness extension and asymmetric shift of the midsurface. Upon linearization with zero Cosserat couple modulus μ_c = 0, one recovers the infinitesimal-displacement Reissner–Mindlin model. It is shown that the Cosserat shell formulation admits minimizers even for μ_c = 0, in which case the drill-energy is absent. The midsurface deformation m is found in H¹(ω, ℝ³). Since the existence of energy minimizers rather than equilibrium solutions is established, the proposed analysis includes the large deformation/large rotation buckling behaviour of thin shells.

In this paper we study a coupled nonlinear Schrödinger–Poisson problem with radial functions. This system has been introduced as a model describing standing waves for the nonlinear Schrödinger equations in the presence of the electrostatic field.

We provide necessary conditions for concentration on sphere for the solutions of this kind of problem extending the results already known.

The Schrödinger–Poisson system describes standing waves for the nonlinear Schrödinger equation interacting with the electrostatic field.

We deal with the semiclassical states for this system and prove the existence of radial solutions concentrating on spheres in the presence of an external potential and with a non-constant density charge.

In particular, we show that the necessary conditions obtained in Part I are also sufficient if suitable non-degeneracy conditions are assumed.

We use a perturbation technique in a variational setting.

We study a new class of electromagnetostatic problems in the variational framework of the subspace of W^1,p(Ω) of vector functions with zero divergence and zero normal trace, for , in smooth, bounded and simply connected domains Ω of ℝ³. We prove a Poincaré–Friedrichs type inequality and we obtain the existence of steady-state solutions for an electromagnetic induction heating problem and for a quasi-variational inequality modelling a critical state generalized problem for type-II superconductors.

This paper deals with fracture mechanics in periodically perforated domains. Our aim is to provide a variational model for brittle porous media in the case of anti-planar elasticity.

Given the perforated domain Ω_ε ⊂ ℝ^N (ε being an internal scale representing the size of the periodically distributed perforations), we will consider a total energy of the type

We study the asymptotic behavior of the functionals in terms of Γ-convergence as ε → 0. As a first (nontrivial) step we show that the domain of any limit functional is SBV(Ω) despite the degeneracies introduced by the perforations. Then we provide explicit formula for the bulk and surface energy densities of the Γ-limit, representing in our model the effective elastic and brittle properties of the porous medium, respectively.