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.

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.

Nonexistence of nodal solutions of nonlinear elliptic equations

2001 ◽  
Vol 46 (8) ◽  
pp. 1111-1122 ◽  
Author(s):  
Soohyun Bae ◽  
Dae Hyeon Pahk
Keyword(s):  
Elliptic Equations ◽  
Nonlinear Elliptic Equations ◽  
Nodal Solutions ◽  
Nonlinear Elliptic

Nodal solutions for some nonlinear elliptic equations

2007 ◽  
Vol 186 (1) ◽  
pp. 589-597 ◽  
Author(s):  
Anouar Ben Mabrouk ◽  
Mohamed Lakdar Ben Mohamed
Keyword(s):  
Elliptic Equations ◽  
Nonlinear Elliptic Equations ◽  
Nodal Solutions ◽  
Nonlinear Elliptic

scholarly journals Positive Solutions of Nonlinear Elliptic Equations with Nonlinear Boundary Conditions

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Hua Luo
Keyword(s):  
Boundary Conditions ◽  
Positive Solutions ◽  
Elliptic Equations ◽  
Nonlinear Boundary ◽  
Eigenvalue Problems ◽  
Nonlinear Elliptic Equations ◽  
Nonlinear Boundary Conditions ◽  
Nonlinear Elliptic ◽  
Elliptic Eigenvalue Problems ◽  
Bifurcation From Interval

This paper discusses bifurcation from interval for the elliptic eigenvalue problems with nonlinear boundary conditions and studies the behavior of the bifurcation components.

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