System Of Nonlinear Elliptic Equations Research Articles

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.

A priori estimates and existence of solutions to a system of nonlinear elliptic equations

We consider the following nonlinear elliptic equations−Δu+u+λϕ(x)|u|r−2u=|u|p−1u,x∈Ω,−Δϕ(x)=|u|q,x∈Ω,ϕ(x)=u(x)=0,x∈∂Ω,(P)where p,q,r>1, λ is a parameter and Ω⊂R3 is a bounded domain. For q=r=2, the equations reduce to the Schrödinger–Poisson equations. Without the need of imposing constraint that q must be equal to r, we establish a priori estimates, the nonexistence and existence of solutions for problem (P). Our results extend previous work for the case q=r to more general case.

Existence and multiplicity of triple weak solutions for a nonlinear elliptic problem with fourth-order operator and Hardy potential

<abstract><p>This study investigates the existence of triple weak solutions for a system of nonlinear elliptic equations with a fourth-order operator. The problem arises in the mathematical modeling of complex physical phenomena.</p></abstract>

Constructive proofs for localised radial solutions of semilinear elliptic systems on

Ground state solutions of elliptic problems have been analysed extensively in the theory of partial differential equations, as they represent fundamental spatial patterns in many model equations. While the results for scalar equations, as well as certain specific classes of elliptic systems, are comprehensive, much less is known about these localised solutions in generic systems of nonlinear elliptic equations. In this paper we present a general method to prove constructively the existence of localised radially symmetric solutions of elliptic systems on . Such solutions are essentially described by systems of non-autonomous ordinary differential equations. We study these systems using dynamical systems theory and computer-assisted proof techniques, combining a suitably chosen Lyapunov–Perron operator with a Newton–Kantorovich type theorem. We demonstrate the power of this methodology by proving specific localised radial solutions of the cubic Klein–Gordon equation on , the Swift–Hohenberg equation on , and a three-component FitzHugh–Nagumo system on . These results illustrate that ground state solutions in a wide range of elliptic systems are tractable through constructive proofs.

Existence of infinitely many positive radial solutions for an iterative system of nonlinear elliptic equations on an exterior domain

Existence of infinitely many positive radial solutions for an iterative system of nonlinear elliptic equations on an exterior domain

Positive Solutions for an Iterative System of Nonlinear Elliptic Equations

Positive Solutions for an Iterative System of Nonlinear Elliptic Equations

A new algorithm to determine the creation or depletion term of parabolic equations from boundary measurements

We propose a robust numerical method to find the coefficient of the creation or depletion term of parabolic equations from the measurement of the lateral Cauchy information of their solutions. Most papers in the field study this nonlinear and severely ill-posed problem using optimal control. The main drawback of this widely used approach is the need of some advanced knowledge of the true solution. In this paper, we propose a new method that opens a door to solve nonlinear inverse problems for parabolic equations without any initial guess of the true coefficient. This claim is confirmed numerically. The key point of the method is to derive a system of nonlinear elliptic equations for the Fourier coefficients of the solution to the governing equation with respect to a special basis of L2. We then solve this system by a predictor–corrector process, in which our computation to obtain the first and second predictors is effective. The desired solution to the inverse problem under consideration follows.

A parameter robust numerical method for a nonlinear system of singularly perturbed elliptic equations

This paper deals with a parameter robust numerical method for solving a nonlinear singularly perturbed reaction–diffusion system of elliptic equations. Existence and uniqueness of a solution to the nonlinear system of elliptic equations are presented. Existence and uniqueness of a solution to a nonlinear difference scheme, which approximates the nonlinear elliptic system, are established. The uniform convergence of the nonlinear difference scheme on piecewise uniform and log-meshes to the solution of the nonlinear elliptic system is established. A monotone iterative method for solving the nonlinear difference scheme is given. The uniform convergence of monotone iterates to the solutions of the nonlinear difference scheme and to the nonlinear elliptic system is proved. Numerical experiments confirm the established theoretical results.

The regularity transformation equations: An elliptic mechanism for smoothing gravitational metrics in general relativity

A central question in General Relativity (GR) is how to determine whether singularities are geometrical properties of spacetime, or simply anomalies of a coordinate system used to parameterize the spacetime. In particular, it is an open problem whether there always exist coordinate transformations which smooth a gravitational metric to optimal regularity, two full derivatives above the curvature tensor, or whether regularity singularities exist. We resolve this open problem above a threshold level of smoothness by proving in this paper that the existence of such coordinate transformations is equivalent to solving a system of nonlinear elliptic equations in the unknown Jacobian and transformed connection, both viewed as matrix valued differential forms. We name these the Regularity Transformation equations, or RT-equations. In a companion paper we prove existence of solutions to the RT-equations for connections $\Gamma\in W^{m,p},$ curvature ${\rm Riem}(\Gamma) \in W^{m,p}$, assuming $m\geq1$, $p>n$. Taken together, these results imply that there always exist coordinate transformations which smooth arbitrary connections to optimal regularity, (one derivative more regular than the curvature), and there are no regularity singularities, above the threshold $m\geq1$, $p>n$. Authors are currently working on extending these methods to the case of GR shock waves, when gravitational metrics are only Lipschitz continuous, ($m=0$, $p=\infty$), and optimal regularity is required to recover basic properties of spacetime.

Optimal metric regularity in general relativity follows from the RT-equations by elliptic regularity theory in $L^p$ -spaces

Shock wave solutions of the Einstein equations have been constructed in coordinate systems in which the gravitational metric is only Lipschitz continuous, but the connection $\Gamma$ and curvature $Riem(\Gamma)$ are both in $L^{\infty}$. At this low level of regularity, the physical meaning of such gravitational metrics remains problematic. Here we address the mathematical problem as to whether the condition that $Riem(\Gamma)$ has the same regularity as $\Gamma$, is sufficient for the existence of a coordinate transformation which perfectly cancels out the jumps in the leading order derivatives of $\delta\Gamma$, thereby raising the regularity of the connection and the metric by one order--a subtle problem. We have now discovered, in a framework much more general than GR shock waves, that the regularization of non-optimal connections is determined by a nonlinear system of elliptic equations with matrix valued differential forms as unknowns, the Regularity Transformation equations, or RT-equations. In this paper we establish the first existence theory for the nonlinear RT-equations in the general case when $\Gamma, {\rm Riem}(\Gamma)\in W^{m,p}$, $m\geq1$, $n<p< \infty$, where $\Gamma$ is any affine connection on an $n$-dimensional manifold. From this we conclude that for any such connection $\Gamma(x) \in W^{m,p}$ with ${\rm Riem}(\Gamma) \in W^{m,p}$, $m\geq1$, $n<p< \infty$, given in $x$-coordinates, there always exists a coordinate transformation $x\to y$ such that $\Gamma(y) \in W^{m+1,p}$. That is, $\Gamma$ exhibits optimal regularity in $y$-coordinates. The problem of optimal regularity for the hyperbolic Einstein equations is thus resolved by elliptic regularity theory in $L^p$-spaces applied to the RT-equations.

Classification of Solutions to a Critically Nonlinear System of Elliptic Equations on Euclidean Half-Space

For N � 3 and non-negative real numbers aij and bij (i, j = 1,··· , m), the semi-linear elliptic system ( �u i + Q m=1 u aij j = 0 in R N @ui @yN = ci Q m j=1 u bij j on @R N i = 1,··· , m is considered, where R N is the upper half of N-dimensional Euclidean space. Under suitable assumptions on the exponents aij and bij, a classification theorem for the positive C 2 (R N) C 1 (R N )-solutions of this system is proven.

Existence theorems for non-Abelian Chern–Simons–Higgs vortices with flavor

In this paper we establish the existence of vortex solutions for a Chern–Simons–Higgs model with gauge group SU(N)×U(1) and flavor SU(N). These symmetries ensure the existence of genuine non-Abelian vortices through a color–flavor locking. Under a suitable ansatz we reduce the problem to a 2×2 system of nonlinear elliptic equations with exponential terms. We study this system over the full plane and over a doubly periodic domain, respectively. For the planar case we use a variational argument to establish the existence result and derive the decay estimates of the solutions. Over the doubly periodic domain we show that the system admits at least two gauge-distinct solutions carrying the same physical energy by using a constrained minimization approach and the mountain-pass theorem. In both cases we get the quantized vortex magnetic fluxes and electric charges.

Systems of elliptic equations involving multiple critical nonlinearities and different Hardy-type terms in [formula omitted

In this paper, we study nonlinear systems of elliptic equations involving different Hardy-type terms and multiple critical nonlinearities. By very technical variational methods, local compactness of Palais–Smale sequences is established and values of the best Sobolev-type constants are compared. Finally, the existence of minimizers to related Rayleigh quotient is verified and the existence of positive solutions to elliptic systems is thus proved.

Multiple Vortices in the Aharony–Bergman–Jafferis–Maldacena Model

Vortices in non-Abelian gauge field theory play important roles in confinement mechanism and are governed by systems of nonlinear elliptic equations of complicated structures. In this paper, we present a series of existence and uniqueness theorems for multiple vortex solutions of the BPS vortex equations, arising in the dual-layered Chern–Simons field theory developed by Aharony, Bergman, Jafferis, and Maldacena, over \({\mathbb{R}^2}\) and on a doubly periodic domain. In the full-plane setting, we show that the solution realizing a prescribed distribution of vortices exists and is unique. In the compact setting, we show that a solution realizing n prescribed vortices exists over a doubly periodic domain \({\Omega}\) if and only if the condition $$n < \frac{\lambda |\Omega|}{2 \pi}$$holds, where \({\lambda >0 }\) is the Higgs coupling constant. In this case, if a solution exists, it must be unique. Our methods are based on calculus of variations.

Existence of multiple vortices in supersymmetric gauge field theory

Two sharp existence and uniqueness theorems are presented for solutions of multiple vortices arising in a six-dimensional brane-world supersymmetric gauge field theory under the general gauge symmetry group G = U (1)× SU ( N ) and with N Higgs scalar fields in the fundamental representation of G . Specifically, when the space of extra dimension is compact so that vortices are hosted in a 2-torus of volume | Ω |, the existence of a unique multiple vortex solution representing n 1 ,…, n N , respectively, prescribed vortices arising in the N species of the Higgs fields is established under the explicitly stated necessary and sufficient condition where e and g are the U (1) electromagnetic and SU ( N ) chromatic coupling constants, v measures the energy scale of broken symmetry and is the total vortex number; when the space of extra dimension is the full plane, the existence and uniqueness of an arbitrarily prescribed n -vortex solution of finite energy is always ensured. These vortices are governed by a system of nonlinear elliptic equations, which may be reformulated to allow a variational structure. Proofs of existence are then developed using the methods of calculus of variations.

Existence of entire positive solutions for a critical system of nonlinear elliptic equations

In this paper, using the fibering method introduced by Pohozaev, we establish an existence of multiple nontrivial positive solutions for a system of nonlinear elliptic equations in R N with lack of compactness studying the properties of Palais–Smale sequence of the energy functional associated with the system.

Sharp existence and uniqueness theorems for non-Abelian multiple vortex solutions

Vortices in non-Abelian gauge field theory play essential roles in the mechanism of color confinement and are governed by systems of nonlinear elliptic equations of complicated structure. In this paper, we present a series of sharp existence and uniqueness theorems for multiple vortex solutions of the non-Abelian BPS equations over R 2 and on a doubly periodic domain. Our methods are based on calculus of variations which may be used to analyze more extended problems. The necessary and sufficient conditions for the existence of a unique solution in the doubly periodic situation are expressed in terms of physical parameters involved explicitly.

Numerical solutions of coupled systems of nonlinear elliptic equations

AbstractThis article deals with numerical solutions of a general class of coupled nonlinear elliptic equations. Using the method of upper and lower solutions, monotone sequences are constructed for difference schemes which approximate coupled systems of nonlinear elliptic equations. This monotone convergence leads to existence‐uniqueness theorems for solutions to problems with reaction functions of quasi‐monotone nondecreasing, quasi‐monotone nonincreasing and mixed quasi‐monotone types. A monotone domain decomposition algorithm which combines the monotone approach and an iterative domain decomposition method based on the Schwarz alternating, is proposed. An application to a reaction‐diffusion model in chemical engineering is given. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 621–640, 2012

New stationary distributions of the Vlasov–Maxwell–Fokker–Planck's system

In this paper we investigate the nonlinear Vlasov–Maxwell–Fokker–Plank (VMFP) system. We propose new families of distributions for the stationary VMFP equation, which is reduced to certain systems of nonlinear elliptic equations depending on arbitrary parameters. We consider the cases when these elliptic systems allow the reduction to the nonlinear PDEs of fourth order, or to the well-known Liouville equation.

A new approach to interior regularity of elliptic systems with quadratic Jacobian structure in dimension two

We prove a new regularity result for systems of nonlinear elliptic equations with quadratic Jacobian type nonlinearity in dimension two. Our proof is based on an adaptation of John Lewis' method which has not been used for such systems so far.