We prove the existence of multiple positive solutions for elliptic systems with linear boundary conditions of Neumann type. We suppose that the nonlinearities grow quadratically with respect to gradient. A key step is to obtain a priori bound on the derivatives by using a Gronwall-type inequality. Our approach is topological and relies on the fixed point index.

We show how the Poincaré-Birkhoff theorem for Hamiltonian systems can be used to find multiple solutions of the antiperiodic problem. Applications are given to scalar second order differential equations whose nonlinearities provide a twist in the phase plane, among which those with a superlinear or sublinear behaviour at infinity.

We present a class of nonlinear mappings, properly containing the nonexpansive ones, enjoying the fixed point property in orthogonally convex Banach spaces.

<jats:p>We investigate a relation between the Harnack inequalities and the (a priori) growth estimates for positive solutions of quasilinear elliptic equations with nonlinear terms involving the solution and its gradient in an arbitrary domain in \(\mathbb{R}^N\).</jats:p>

<jats:p>The analytic and formal solutions to a family of singularly perturbed partial differential equations in the complex domain involving two complex time variables are considered. The analytic continuation properties of the solution of an auxiliary problem in the Borel plane overcomes the absence of adequate domains which would guarantee summability of the formal solution. Moreover, several exponential decay rates of the difference of analytic solutions with respect to the perturbation parameter at the origin are observed, leading to several asymptotic levels relating the analytic and the formal solution.</jats:p>

The bound states and decay time in a certain quantum well structure (GaMnAs/GaAs) were analyzed and identified at the minimum decay time. Through the analysis of quantum mathematical equations, we derived specific formulas for energies that significantly amplify in the numerical solutions of equations throughout all dimensions of confinement. The quantification and barriers, alongside the well width, without altering the parameters utilized, were predominantly influenced by the spatial dimension parameters, such as the barrier height and well width. The principal bound state and lowest decay time were determined at a well width of 40 Å and a barrier thickness of 46.27 Å. This work revealed a novel characteristic known as interfacial tunnelling, which refers to the phenomenon where an electron establishes a tunnelling state between two interfaces. This tunnelling process is significantly influenced by the characteristics of the materials used, as well as the dimensions of the wells and barriers.

Understanding the local evolution of phase transformations in steels, particularly the γ (austenite) → α (ferrite) transformation, is crucial for controlling the microstructure and properties of steel components. Over recent decades, significant progress has been made in the numerical modeling of this complex phenomenon. This development has been driven by both scientific curiosity and industrial needs, especially in processes such as hot rolling, forging, thermal treatment, etc. The developed models have evolved from simple solutions based on local equilibrium to more complex approaches that consider local heterogeneities. Modern computational approaches, such as Phase-Field (PF), Level-Set (LS), Cellular Automata (CA), Monte Carlo (MC) or Vertex based simulations, allow for the precise reproduction of microstructural evolution considering local instabilities. They also enable the analysis of phase boundary motion in an explicit manner. These techniques also allow for direct integration with thermodynamic data and mechanical models, thereby better capturing the physical mechanisms of phase transformations, such as chemical composition, diffusion resistance, or the influence of deformation. An overview of the state of the art in this area is presented within the paper. The model’s concepts, assumptions, fundamental equations, advantages, limitations, and potential practical applications are summarized. Special attention is given to modeling the γ → α transformation by the Cellular Automata method. The importance of incorporating phenomena such as diffusion, nucleation, and growth is emphasized. The need for consistency between experimental results and simulations is also highlighted.

In this paper, we study the existence of a nonnegative weak solution to the following nonlocal variational inequality: \[\int_{\mathbb{R}^N}(-\Delta)^{\frac{s}{2}} u (-\Delta)^{{\frac{s}{2}}}(v-u)dx+\int_{\mathbb{R}^N}(1+\lambda M(x))u(v-u)dx \geq \int_{\mathbb{R}^N}f(u)(v-u)dx, \] for all \(v \in\mathbb{K}\), where \(s\in (0,1)\) and \(M\) is a continuous steep potential well on \(\mathbb{R}^N\). Using penalization techniques from del Pino and Felmer, as well as from Bensoussan and Lions, we establish the existence of nonnegative weak solutions. These solutions localize near the potential well \(\operatorname{Int}(M^{-1}(0))\).

<jats:p>This paper introduces a methodology to derive explicit power series approximations for the limit cycle periodic solutions of the Hopf bifurcation in autonomous discrete delay differential equations (DDE). The procedure extends the methodology introduced by Casal and Freedman in 1980, by providing a detailed algorithm that iteratively performs systematic calculations up to any desired order of approximation, ensuring a specific error tolerance for any nonlinear DDE presenting a Hopf bifurcation. The methodology is applied to three relevant delay-differential models to illustrate its features: a recently introduced car-following mobility model that explains oscillations in road traffic, a SIR epidemic model for propagation of diseases with temporary immunity, and a simplified macroeconomic system to model business cycles.</jats:p>

<jats:p>In this paper, we investigate some classes of Neumann fractional \(p\)-Laplacian problems. We prove the existence and multiplicity of nontrivial solutions for several different nonlinearities, by using variational methods and critical point theory based on cohomological linking.</jats:p>