scholarly journals On the normalized eigenvalue problems for nonlinear elliptic operators

2007 ◽  
Vol 329 (1) ◽  
pp. 51-64 ◽  
Author(s):  
Jing Lin
Keyword(s):  
Eigenvalue Problems ◽  
Elliptic Operators ◽  
Nonlinear Elliptic

scholarly journals Generalized principal eigenvalues of convex nonlinear elliptic operators in ℝN

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Anup Biswas ◽  
Prasun Roychowdhury
Keyword(s):  
Eigenvalue Problem ◽  
Principal Eigenvalue ◽  
Unbounded Domains ◽  
Elliptic Operators ◽  
Maximum Principles ◽  
Generalized Eigenvalues ◽  
Nonlinear Elliptic ◽  
Generalized Eigenvalue ◽  
General Convex ◽  
Appl Math

AbstractWe study the generalized eigenvalue problem in {\mathbb{R}^{N}} for a general convex nonlinear elliptic operator which is locally elliptic and positively 1-homogeneous. Generalizing [H. Berestycki and L. Rossi, Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains, Comm. Pure Appl. Math. 68 2015, 6, 1014–1065], we consider three different notions of generalized eigenvalues and compare them. We also discuss the maximum principles and uniqueness of principal eigenfunctions.

On the viscosity solutions of eigenvalue problems for a class of nonlinear elliptic equations

2019 ◽  
Vol 12 (4) ◽  
pp. 393-421
Author(s):  
Tilak Bhattacharya ◽  
Leonardo Marazzi
Keyword(s):  
Viscosity Solutions ◽  
Elliptic Equations ◽  
Eigenvalue Problems ◽  
A Priori ◽  
Nonlinear Elliptic Equations ◽  
First Eigenvalue ◽  
Nonlinear Elliptic ◽  
Degenerate Elliptic ◽  
First Eigenfunction ◽  
Nonlinear Degenerate

AbstractWe consider viscosity solutions of a class of nonlinear degenerate elliptic equations, involving a parameter, on bounded domains. These arise in the study of eigenvalue problems. We prove comparison principles and a priori supremum bounds for the solutions. We also address the eigenvalue problem and, in many instances, show the existence of the first eigenvalue and an associated positive first eigenfunction.

Extremal, Nodal and Stable Solutions for Nonlinear Elliptic Equations

2015 ◽  
Vol 15 (3) ◽  
Author(s):  
Leszek Gasiński ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Eigenvalue Problems ◽  
Principal Eigenvalue ◽  
Nonlinear Elliptic Equations ◽  
Nodal Solutions ◽  
The Past ◽  
Nonlinear Elliptic ◽  
Multiplicity Theorem ◽  
The Stability ◽  
Laplacian Equations ◽  
Special Case

AbstractWe consider Dirichlet p-Laplacian equations which may be resonant with respect to the principal eigenvalue at ±∞. We show the existence of extremal nontrivial constant sign solutions and of nodal solutions. In the semilinear case (p = 2) we produce additional nodal solutions. We show that certain parametric equations (eigenvalue problems) studied in the past, are a special case of our multiplicity theorem. Finally, we establish the stability of the extremal solutions.

MULTIGRID-CONJUGATE GRADIENT TYPE METHODS FOR REACTION–DIFFUSION SYSTEMS

2004 ◽  
Vol 14 (10) ◽  
pp. 3587-3605 ◽  
Author(s):  
S.-L. CHANG ◽  
H.-S. CHEN ◽  
C.-S. CHIEN
Keyword(s):  
Eigenvalue Problems ◽  
Singular Systems ◽  
Reaction Diffusion ◽  
Continuation Methods ◽  
Approximation Schemes ◽  
Reaction Diffusion Systems ◽  
Nonlinear Elliptic ◽  
Diffusion Systems ◽  
Relaxation Scheme ◽  
Gradient Type

We study multigrid methods in the context of continuation methods for reaction–diffusion systems, where the Bi-CGSTAB and GMRES methods are used as the relaxation scheme for the V-cycle, W-cycle and full approximation schemes, respectively. In particular, we apply the results of Brown and Walker [1997] to investigate how the GMRES method can be used to solve nearly singular systems that occur in continuation problems. We show that for the sake of switching branches safely, one would rather solve a perturbed problem near bifurcation points. We propose several multigrid-continuation algorithms for curve-tracking in nonlinear elliptic eigenvalue problems. Our numerical results show that the algorithms proposed have the advantage of being robust and easy to implement.

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