A uniqueness principle for up ≤ (−Δ)α 2u in the Euclidean space

2016 ◽  
Vol 18 (06) ◽  
pp. 1650019 ◽  
Author(s):  
Y. Wang ◽  
J. Xiao
Keyword(s):  
Weak Solution ◽  
Euclidean Space ◽  
Fractional Order ◽  
Elliptic Equations ◽  
Differential Inequality ◽  
Nonlinear Elliptic Equations ◽  
Local Behavior ◽  
Nonlinear Elliptic ◽  
Global And Local ◽  
Appl Math

This paper establishes such a uniqueness principle that under [Formula: see text] the fractional order differential inequality [Formula: see text] has the property that if [Formula: see text] then a non-negative weak solution to [Formula: see text] is unique, and if [Formula: see text] then the uniqueness of a non-negative weak solution to [Formula: see text] occurs when and only when [Formula: see text], thereby innovatively generalizing Gidas–Spruck’s result for [Formula: see text] in [Formula: see text] discovered in [B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math. 34 (1981) 525–598].

scholarly journals A Class of Degenerate Nonlinear Elliptic Equations in Weighted Sobolev Space

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Rasmita Kar
Keyword(s):  
Weak Solution ◽  
Sobolev Space ◽  
Boundary Value Problem ◽  
Elliptic Equations ◽  
Dirichlet Boundary ◽  
Weighted Sobolev Space ◽  
Nonlinear Elliptic Equations ◽  
Dirichlet Boundary Value Problem ◽  
Nonlinear Elliptic ◽  
Bounded Nonlinearity

We prove the existence of a weak solution for the degenerate nonlinear elliptic Dirichlet boundary-value problem Lu-μug1+hu,∇ug2=f in Ω, u=0 on ∂Ω, in a suitable weighted Sobolev space, where Ω⊂ℝn is a bounded domain and h is a continuous bounded nonlinearity.

scholarly journals Uniqueness of weak solution for nonlinear elliptic equations in divergence form

2000 ◽  
Vol 23 (5) ◽  
pp. 313-318 ◽  
Author(s):  
Xu Zhang
Keyword(s):  
Weak Solution ◽  
Weak Solutions ◽  
Elliptic Equations ◽  
Nonlinear Elliptic Equations ◽  
Divergence Form ◽  
Uniqueness Result ◽  
Quasilinear Elliptic Equations ◽  
Nonlinear Elliptic ◽  
Quasilinear Elliptic

We study the uniqueness of weak solutions for quasilinear elliptic equations in divergence form. Some counterexamples are given to show that our uniqueness result cannot be improved in the general case.

scholarly journals On a Class of Anisotropic Nonlinear Elliptic Equations with Variable Exponent

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Guoqing Zhang ◽  
Hongtao Zhang
Keyword(s):  
Weak Solution ◽  
Elliptic Equations ◽  
Existence And Uniqueness ◽  
Variable Exponent ◽  
Nonlinear Elliptic Equations ◽  
Nonlinear Elliptic ◽  
Priori Estimates

Based on truncation technique and priori estimates, we prove the existence and uniqueness of weak solution for a class of anisotropic nonlinear elliptic equations with variable exponentp(x)→growth. Furthermore, we also obtain that the weak solution is locally bounded and regular; that is, the weak solution isLloc∞(Ω)andC1,α(Ω).

scholarly journals An existence and approximation theorem for solutions of degenerate nonlinear elliptic equations

2018 ◽  
Vol 72 (1) ◽  
pp. 29
Author(s):  
Albo Carlos Cavalheiro
Keyword(s):  
Weak Solution ◽  
Elliptic Equations ◽  
Approximation Theorem ◽  
Nonlinear Elliptic Equations ◽  
Nonlinear Elliptic

The main result establishes that a weak solution of degenerate nonlinear  elliptic equations can be approximated by a sequence of solutions for non-degenerate nonlinear elliptic equations.

A new concept of reduced measure for nonlinear elliptic equations

2004 ◽  
Vol 339 (3) ◽  
pp. 169-174 ◽  
Author(s):  
Haïm Brezis ◽  
Moshe Marcus ◽  
Augusto C. Ponce
Keyword(s):  
Elliptic Equations ◽  
Nonlinear Elliptic Equations ◽  
Nonlinear Elliptic
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