Nonlinear Elliptic Equations Research Articles

An optimal transport approach for 3D electrical impedance tomography

Abstract This work solves the three-dimensional inverse boundary value problem with the quadratic Wasserstein distance (W2), which originates from the optimal transportation (OT) theory. The computation of the W2 distance on the manifold surface is boiled down to solving the generalized Monge-Ampère equation, whose solution is directly related to the gradient of the W2 distance. An efficient first-order method based on iteratively solving Poisson's equation is introduced to solve the fully nonlinear elliptic equation. Combining with the adjoint-state technique, the optimization framework based on the W2 distance is developed to solve the three-dimensional electrical impedance tomography (EIT) problem. The proposed method is especially suitable for severely ill-posed and highly nonlinear inverse problems. Numerical experiments demonstrate that our method improves the stability and outperforms the traditional regularization methods.

A PNP ion channel deep learning solver with local neural network and finite element input data

Abstract This paper presents a deep learning method for solving an improved one-dimensional Poisson–Nernst–Planck ion channel (PNPic) model, called the PNPic deep learning solver. The solver combines a novel local neural network, adapted from the neural network with local converging inputs, with an efficient PNPic finite element solver, developed in this work. In particular, the local neural network is extended to handle the complexities of the PNPic model—a system of nonlinear convection–diffusion and elliptic equations with multiple subdomains connected by interface conditions. The PNPic finite element solver efficiently generates input and reference datasets for fast training the local neural network, as well as input datasets for quickly predicting PNPic solutions with high accuracy for a family of PNPic models. Initial numerical tests, involving perturbations of model parameters and interface locations, demonstrate that the PNPic deep learning solver can generate highly accurate numerical solutions.

Analysis of a nonlinear elliptic system with variable coefficients and Hardy potential with Carathéodory nonlinearity: Existence

Abstract This study investigates existence result of a nontrivial weak solutions for a system of nonlinear elliptic equations involving Hardy Potentials. The problem arises in the mathematical modeling of complex physical phenomena.

Nonlinear Elliptic Equations with Variable Exponents Anisotropic Sobolev Weights and Natural Growth Terms

Abstract The purpose of our paper is to prove the existence of the distributional solutions for anisotropic nonlinear elliptic equations with variable exponents, which contain lower order terms dependent on the gradient of the solution and on the solution itself. The terms are weighted, and the main results rely on the possibility of comparing the weights with each other, where the right-hand side is a sum of the natural growth term and the datum f ∈ L 1(Ω). Furthermore the weight function θ(·) is in W̊ 1,p→(·) (Ω), with θ(·) > 0 and connected with the coefficient b(·) ∈ L 1(Ω) of the lower order term.

Застосування методу двобічних наближень до знаходження додатних аксіально-симетричних розв’язків крайових задач з монотонними нелінійностями

This paper considers the application of the method of two-sided approximations for finding positive solutions to boundary value problems for nonlinear elliptic differential equations with axial symmetry. The case of Dirichlet boundary conditions (first kind) is considered, with a power-type monotone nonlinearity, where the exponent ranges from 0 to 1. By transitioning to polar coordinates in the boundary value problem for the elliptic equation, due to the axial symmetry of the solution, the problem is reduced to a boundary value problem for an ordinary differential equation on an interval with respect to a function that depends only on the polar radius, thus eliminating the dependence on the angle. The pole of the polar coordinate system becomes a singular point, where a boundedness condition must be imposed. For the boundary value problem, a Green's function is constructed to further reduce the problem to a Hammerstein integral equation. The integral equation is treated as a nonlinear operator equation in a Banach space of continuous functions on the interval, partially ordered by a cone of non-negative continuous functions on this interval. The operator is examined for properties such as monotonicity (isotonicity), positivity, boundedness, and concavity. Next, the initial approximation is found as the endpoints of the invariant conical segment for the isotone operator in such a way as to ensure the highest convergence rate of the iterative process. The following iterative sequences of two-sided approximations are constructed: the first sequence, which is non-decreasing with respect to the cone, and the second sequence, which is non-increasing with respect to the cone. At each iteration, the arithmetic mean of the upper and lower approximations is taken as the next approximation. The iterative process is terminated when the error estimate of the solution satisfies the specified accuracy. The theoretical results obtained in this work were validated through a computational experiment. The dependence of the solution on the parameters in the right-hand side was analyzed and illustrated with corresponding graphs

A novel multilevel finite element method for a generalized nonlinear Schrödinger equation

In this article, we focus on an efficient multilevel finite element method to solve the time-dependent nonlinear Schrödinger equation which is one of the most important equations of mathematical physics. For the time derivative, we adopt implicit schemes including the backward Euler method and the Crank–Nicolson method. Based on these stable implicit schemes, the proposed method requires solving a nonlinear elliptic problem at each time step. For these nonlinear elliptic equations, a multilevel mesh sequence is constructed. At each mesh level, we first derive a rough approximation by correcting the approximation of the previous mesh level in a special correction subspace. The correction subspace is composed of a coarse finite element space and an additional approximate solution derived from the previous mesh level. Next, we only need to solve a linearized elliptic equation by inserting the rough approximation into the nonlinear term. Then, we derive an accurate approximate solution by performing the aforementioned solving process on the multilevel mesh sequence until we reach the final mesh level. Owing to the special construct of the correction subspace, we derive a multilevel finite element method to solve the nonlinear Schrödinger equation for the first time, and meanwhile we also derive an optimal error estimate with linear computational complexity. Additionally, unlike the existing multilevel methods for nonlinear problems, that typically require bounded second-order derivatives of the nonlinear terms, the nonlinear term in our study requires only one-order derivatives. Numerical results are provided to support our theoretical analysis and demonstrate the efficiency of the presented method.

Global Solution for a Fourth-Order Nonlinear Elliptic Equation

Objective: In this work, the existence of a global solution for a fourth-order nonlinear elliptic equation with boundary conditions is studied. Method: Variational methods are applied to the proposed model, naturally using the Banach Fixed Point Contraction Theorem. Results and Discussion: The conclusion of the result obtained is that for the nonlinear elliptic equation of the fourth order they have equivalent norms in the conveniently defined spaces. On the other hand, the existence of a solution exists whenever Banach's fixed-point theorem can be applied. Implications of the research: Apply in the first university years the application of global solution for a fourth-order nonlinear elliptic equation in all university careers and specialties as part of the management of the tools of Information and Communication Technology (TIC). Originality/Value: This paper also provides an alternative approach to Analysis of global solution for a fourth-order nonlinear elliptic equation.

Positive solutions of nonlinear elliptic equations involving unbounded variable exponents and convection term

Positive solutions of nonlinear elliptic equations involving unbounded variable exponents and convection term

Radially symmetric solutions of a nonlinear singular elliptic equation

Radially symmetric solutions of a nonlinear singular elliptic equation

W2,p-Regularity of Lp Viscosity Solutions to Fully Nonlinear Elliptic Equations with Low-Order Terms

In this paper, we consider the following fully nonlinear elliptic equation F(D2u, Du, x) = f(x), where the operator F satisfies structure condition and the gradient of solution has Lploc growth rate particularly. We employ the technique from geometric tangential analysis whose basic principle is to transfer the good regularity of the recession operator to the original F by approximation methods and establish a prior local W2,p estimates for - Lp-viscosity solutions to the above equation. Mathematics Subject classification (2010): 35B45; 35R05; 35B65.

On the Sub and Supersolution Method for Nonlinear Elliptic Equations with a Convective Term, in Orlicz Spaces

In this note we provide an overview of some existence (with sign information) and regularity results for differential equations, in which the method of sub and supersolutions plays an important role. We list some classical results and then we focus on the Dirichlet problem, for problems driven by a general differential operator, depending on (x,u,∇u), and with a convective term f. Our framework is that of Orlicz–Sobolev spaces. We also present several examples.

On the critical points of solutions of PDE in non-convex settings: The case of concentrating solutions

In this paper we are concerned with the number of critical points of solutions of nonlinear elliptic equations. We will deal with the case of non-convex, contractile and non-contractile planar domains. We will prove results on the estimate of their number as well as their index. In some cases we will provide the exact calculation. The toy problem concerns the multi-peak solutions of the Gel'fand problem, namely{−Δu=λeu in Ωu=0 on ∂Ω, where Ω⊂R2 is a bounded smooth domain and λ>0 is a small parameter.

Positive solution for a nonlinear elliptic equation on symmetric domains

We show the existence of a positive solution for the Schrodinger quasilinear equation with variable exponents above the critical regime. For that matter, we show an embedding into an Orlicz space of functions modeled over radially symmetric domains. Then we use a Galerkin method combined with a fixed-point argument to obtain a solution. For more information see https://ejde.math.txstate.edu/Volumes/2024/41/abstr.html

Companion-based multi-level finite element method for computing multiple solutions of nonlinear differential equations

The utilization of nonlinear differential equations has resulted in remarkable progress across various scientific domains, including physics, biology, ecology, and quantum mechanics. Nonetheless, obtaining multiple solutions for nonlinear differential equations can pose considerable challenges, particularly when it is difficult to find suitable initial guesses. To address this issue, we propose a pioneering approach known as the Companion-Based Multilevel Finite Element Method (CBMFEM). This novel technique efficiently and accurately generates multiple initial guesses for solving nonlinear elliptic semi-linear equations containing polynomial nonlinear terms through the use of finite element methods with conforming elements. As a theoretical foundation of CBMFEM, we present an appropriate and new concept of the isolated solution to the nonlinear elliptic equations with multiple solutions. The newly introduced concept is used to establish the inf-sup condition for the linearized equation around the isolated solution. Furthermore, it is crucially used to derive a theoretical error analysis of finite element methods for nonlinear elliptic equations with multiple solutions. A number of numerical results obtained using CBMFEM are then presented and compared with a traditional method. These not only show the CBMFEM's superiority, but also support our theoretical analysis. Additionally, these results showcase the effectiveness and potential of our proposed method in tackling the challenges associated with multiple solutions in nonlinear differential equations with different types of boundary conditions.

Existence, Regularity, and Uniqueness of Solutions to Some Noncoercive Nonlinear Elliptic Equations in Unbounded Domains

Existence, Regularity, and Uniqueness of Solutions to Some Noncoercive Nonlinear Elliptic Equations in Unbounded Domains

Rigidity of inverse problems for nonlinear elliptic equations on manifolds

Rigidity of inverse problems for nonlinear elliptic equations on manifolds

Weak solution for biharmonic equation with Navier boundary conditions

The aim of this paper is to study the existence and uniqueness of weak solutions of certain nonlinear second order elliptic partial differential equations.

Energy estimated frequencies of standing-wave solutions to nonlinear Klein-Gordon systems in higher dimensions

A system of nonlinear elliptic equations is considered, where the nonlinear term is the sum of a quadratic form and a Sobolev sub-critical term. An extra assumption is introduced on the nonlinear term, which is minimal among the ones which guarantee the existence of standing-waves obtained by estimating frequencies of minimizing sequences with the energy functional.

Variational nonlinear elliptic equations having large monotonocity in Musielak-Sobolev spaces

Variational nonlinear elliptic equations having large monotonocity in Musielak-Sobolev spaces

Uniqueness of ground states to fractional nonlinear elliptic equations with harmonic potential

In this paper, we prove the uniqueness of ground states to the following fractional nonlinear elliptic equation with harmonic potential, \[ (-\Delta)^s u+ \left(\omega+|x|^2\right) u=|u|^{p-2}u \quad \mbox{in}\ \mathbb{R}^n, \] where $n \geq 1$ , $0< s<1$ , $\omega >-\lambda _{1,s}$ , $2< p< {2n}/{(n-2s)^+}$ , $\lambda _{1,s}>0$ is the lowest eigenvalue of $(-\Delta )^s + |x|^2$ . The fractional Laplacian $(-\Delta )^s$ is characterized as $\mathcal {F}((-\Delta )^{s}u)(\xi )=|\xi |^{2s} \mathcal {F}(u)(\xi )$ for $\xi \in \mathbb {R}^n$ , where $\mathcal {F}$ denotes the Fourier transform. This solves an open question in [M. Stanislavova and A. G. Stefanov. J. Evol. Equ. 21 (2021), 671–697.] concerning the uniqueness of ground states.