Existence and multiplicity analysis of a system of nonlinear elliptic equations: Theoretical results and applications

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

This 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

Projected successive overrelaxation method for finite-element solutions to the Dirichlet problem for a system of nonlinear elliptic equations

We look for solutions of systems of nonlinear elliptic equations with nonlinear boundary conditions and values in some compact convex set M. If the nonlinear terms satisfy a sign condition on the boundary of M and the inhomogeneous terms assume their values in this set existence of solutions is proved.The proof is based on the homotopy invariance of the Leray‐Schauder degree and Weinberger's strong maximum principle.

On a System of Nonlinear Elliptic Equations Arising in Theoretical Physics

One parameter operator imbedding to modify Newton method for solution of nonlinear equations

CANM, a program for numerical solution of a system of nonlinear equations using the continuous analog of Newton's method

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

The Fermi-Amaldi Correction in Spin Polarized Thomas-Fermi Theory

Entire positive solution to the system of quasilinear elliptic equations

Homogenization of the Dirichlet problem for a system of quasilinear elliptic equations in a domain with fine-grained boundary

Homogenization of the Dirichlet problem for a system of quasilinear elliptic equations in a perforated domain

Nonlinear error-in-variable models can advance the development of the systems of additive biomass equations and lead to much higher prediction accuracy of tree biomass than nonlinear seemingly unrelated regression. In this study, the approach of nonlinear error-in-variable models (NEIVM) was compared with nonlinear seemingly unrelated regressions (NSUR) for developing a system of nonlinear additive biomass equations using the data collected in Southern China for Pinus massoniana Lamb. Various tree variables were assessed to explore their contributions to improvement of biomass prediction using the systems of equations. It was found that diameter at breast height (D), total tree height (H) and crown width (CW) significantly contributed to the increase of prediction accuracy. The combinations of D, H, and CW led to three sets of independent variables: (1) D alone; (2) both D and H; and (3) D, H and CW together, which were used for the development of one-predictor, two-predictor and three-predictor systems of biomass equations, respectively. The results showed that both NEIVM and NSUR had high prediction accuracy of biomass for all the systems of biomass equations. For the one-predictor systems of biomass equations, both NEIVM and NSUR led to very similar predictions. However, for the two-predictor and three-predictor systems of biomass equations, the prediction accuracy of NEIVM was much higher than that of NSUR. When the two-predictor system of equations was used, in particular, NEIVM with one-step procedure, that is, by directly partitioning total tree biomass into four basic components, showed a higher accuracy of biomass prediction than NSUR for all the one-predictor, two-predictor and three-predictor systems of equations. This study implies that the NEIVM approach could provide a greater potential to develop a system of biomass equations that are dependent on the predictors with significant measurement errors.

The thermistor problem is modeled as a coupled system of nonlinear elliptic equations. When the conductivity coefficient σ ( u ) \sigma \left ( u \right ) vanishes ( u u = temperature) one of the equations becomes degenerate; this situation is considered in the present paper. We establish the existence of a weak solution and, under some special Dirichlet and Neumann boundary conditions, analyze the structure of the set { σ ( u ) = 0 } \left \{ {\sigma \left ( u \right ) = 0} \right \} and also prove uniqueness.

We study the solvability and the existence of multiple solutions of nonlinear systems of elliptic equations.