Stable and Finite Morse Index Solutions for Dirichlet Problems with Small Diffusion in a Degenerate Case and Problems with Infinite Boundary Values

2009 ◽  
Vol 9 (4) ◽  
Author(s):  
E.N. Dancer
Keyword(s):  
Elliptic Equations ◽  
Morse Index ◽  
Nonlinear Elliptic Equations ◽  
Degenerate Case ◽  
Boundary Values ◽  
Dirichlet Problems ◽  
Small Diffusion ◽  
Weakly Nonlinear ◽  
Nonlinear Elliptic

AbstractWe consider weakly nonlinear elliptic equations with small diffusion in the case where the nonlinearity has a non-nodal zero. We show that there is an unexpected connection with problems with infinite boundary values.

Dirichlet problems for fully anisotropic elliptic equations

2016 ◽  
Vol 147 (1) ◽  
pp. 25-60 ◽  
Author(s):  
Giuseppina Barletta ◽  
Andrea Cianchi
Keyword(s):  
Elliptic Equations ◽  
Partial Differential Operator ◽  
Nonlinear Elliptic Equations ◽  
Dirichlet Problems ◽  
Unified Framework ◽  
Nonlinear Elliptic ◽  
Polynomial Type ◽  
General Convex ◽  
Anisotropic Elliptic Equations ◽  
Relevant Operator

The existence of a non-trivial bounded solution to the Dirichlet problem is established for a class of nonlinear elliptic equations involving a fully anisotropic partial differential operator. The relevant operator depends on the gradient of the unknown through the differential of a general convex function. This function need not be radial, nor have a polynomial-type growth. Besides providing genuinely new conclusions, our result recovers and embraces, in a unified framework, several contributions in the existing literature, and augments them in various special instances.

Fully Nonlinear Elliptic Equations on General Domains

2002 ◽  
Vol 54 (6) ◽  
pp. 1121-1141 ◽  
Author(s):  
Jiguang Bao
Keyword(s):  
Elliptic Equations ◽  
Essential Role ◽  
Nonlinear Elliptic Equations ◽  
Accretive Operator ◽  
Operator Technique ◽  
Dirichlet Problems ◽  
Fully Nonlinear Elliptic Equations ◽  
Fully Nonlinear ◽  
Nonlinear Elliptic ◽  
Generalized Dirichlet

AbstractBy means of the Pucci operator, we construct a function u0, which plays an essential role in our considerations, and give the existence and regularity theorems for the bounded viscosity solutions of the generalized Dirichlet problems of second order fully nonlinear elliptic equations on the general bounded domains, which may be irregular. The approximation method, the accretive operator technique and the Caffarelli's perturbation theory are used.

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