Existence of Solutions with Asymptotic Behavior of Exterior Problems of Fully Nonlinear Uniformly Elliptic Equations

2011 ◽  
Author(s):  
Limei Da
Keyword(s):  
Asymptotic Behavior ◽  
Elliptic Equations ◽  
Existence Of Solutions ◽  
Fully Nonlinear ◽  
Exterior Problems

scholarly journals On the existence of Wp2 solutions for fully nonlinear elliptic equations under either relaxed or no convexity assumptions

2017 ◽  
Vol 19 (06) ◽  
pp. 1750009 ◽  
Author(s):  
N. V. Krylov
Keyword(s):  
Elliptic Equations ◽  
Existence Of Solutions ◽  
Nonlinear Elliptic Equations ◽  
Second Order ◽  
Fully Nonlinear Elliptic Equations ◽  
Sobolev Classes ◽  
Fully Nonlinear ◽  
Nonlinear Elliptic ◽  
Convexity Assumptions ◽  
Second Order Equations

We establish the existence of solutions of fully nonlinear elliptic second-order equations like [Formula: see text] in smooth domains without requiring [Formula: see text] to be convex or concave with respect to the second-order derivatives. Apart from ellipticity nothing is required of [Formula: see text] at points at which [Formula: see text], where [Formula: see text] is any given constant. For large [Formula: see text] some kind of relaxed convexity assumption with respect to [Formula: see text] mixed with a VMO condition with respect to [Formula: see text] are still imposed. The solutions are sought in Sobolev classes. We also establish the solvability without almost any conditions on [Formula: see text], apart from ellipticity, but of a “cut-off” version of the equation [Formula: see text].

scholarly journals Local versus nonlocal elliptic equations: short-long range field interactions

2020 ◽  
Vol 10 (1) ◽  
pp. 895-921
Author(s):  
Daniele Cassani ◽  
Luca Vilasi ◽  
Youjun Wang
Keyword(s):  
Asymptotic Behavior ◽  
Maximum Principle ◽  
Weak Solutions ◽  
Elliptic Equations ◽  
Spectral Properties ◽  
Fractional Laplacian ◽  
Coupling Parameter ◽  
Nonlocal Operators ◽  
The Maximum Principle ◽  
Regularity Results

Abstract In this paper we study a class of one-parameter family of elliptic equations which combines local and nonlocal operators, namely the Laplacian and the fractional Laplacian. We analyze spectral properties, establish the validity of the maximum principle, prove existence, nonexistence, symmetry and regularity results for weak solutions. The asymptotic behavior of weak solutions as the coupling parameter vanishes (which turns the problem into a purely nonlocal one) or goes to infinity (reducing the problem to the classical semilinear Laplace equation) is also investigated.

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