Global gradient estimates for elliptic equations of p(x)-Laplacian type with BMO nonlinearity

2016 ◽  
Vol 2016 (715) ◽  
pp. 1-38 ◽  
Author(s):  
Sun-Sig Byun ◽  
Jihoon Ok ◽  
Seungjin Ryu
Keyword(s):  
Weak Solution ◽  
Elliptic Problem ◽  
Elliptic Equations ◽  
Growth Conditions ◽  
Divergence Form ◽  
Gradient Estimates ◽  
Nonlinear Elliptic Problem ◽  
Nonstandard Growth ◽  
Nonlinear Elliptic ◽  
Nonstandard Growth Conditions

AbstractWe consider a nonlinear elliptic problem in divergence form, with nonstandard growth conditions, on a bounded domain. We obtain the global Calderón–Zygmund type gradient estimates for the weak solution of such a problem in the setting of Lebesgue and Sobolev spaces with variable

Global gradient estimates for asymptotically regular problems of p(x)-Laplacian type

2018 ◽  
Vol 20 (08) ◽  
pp. 1750079 ◽  
Author(s):  
Sun-Sig Byun ◽  
Jehan Oh
Keyword(s):  
Gradient Estimates ◽  
Nonlinear Elliptic Problem ◽  
Optimal Regularity ◽  
Nonstandard Growth ◽  
Nonlinear Elliptic ◽  
Variable Exponent Sobolev Spaces ◽  
General Kind ◽  
Geometric Assumption ◽  
Regular Problems ◽  
Asymptotically Regular

We study an asymptotically regular problem of [Formula: see text]-Laplacian type with discontinuous nonlinearity in a nonsmooth bounded domain. A global Calderón–Zygmund estimate is established for such a nonlinear elliptic problem with nonstandard growth under the assumption that the associated nonlinearity has a more general kind of the asymptotic behavior near the infinity with respect to the gradient variable. We also address an optimal regularity requirement on the nonlinearity as well as a minimal geometric assumption on the boundary of the domain for the nonlinear Calderón–Zygmund theory in the setting of variable exponent Sobolev spaces.

scholarly journals Liouville Theorem for Some Elliptic Equations with Weights and Finite Morse Indices

2016 ◽  
Vol 2016 ◽  
pp. 1-6
Author(s):  
Qiongli Wu ◽  
Liangcai Gan ◽  
Qingfeng Fan
Keyword(s):  
Elliptic Problem ◽  
Elliptic Equations ◽  
Morse Index ◽  
Liouville Theorem ◽  
Bounded Solution ◽  
Positive Parameter ◽  
Nonlinear Elliptic Problem ◽  
Nonlinear Elliptic ◽  
Morse Indices

We establish the nonexistence of solution for the following nonlinear elliptic problem with weights:-Δu=(1+|x|α)|u|p-1uinRN, whereαis a positive parameter. Suppose that1<p<N+2/N-2,α>(N-2)(p+1)/2-NforN≥3orp>1,α>-2forN=2; we will show that this equation does not possess nontrivial bounded solution with finite Morse index.

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.

Existence and non-existence of solutions to elliptic equations with a general convection term

2014 ◽  
Vol 144 (2) ◽  
pp. 225-239 ◽  
Author(s):  
Salomón Alarcón ◽  
Jorge García-Melián ◽  
Alexander Quaas
Keyword(s):  
Elliptic Problem ◽  
Elliptic Equations ◽  
Existence Of Solutions ◽  
Classical Solutions ◽  
Convection Term ◽  
Nonlinear Elliptic Problem ◽  
Smooth Bounded Domain ◽  
Nonlinear Elliptic ◽  
Comparison Arguments ◽  
Increasing Function

In this paper we consider the nonlinear elliptic problem −Δu + αu = g(∣∇u∣) + λh(x) in Ω, u = 0 on ∂Ω, where Ω is a smooth bounded domain of ℝN, α ≥ 0, g is an arbitrary C1 increasing function and h ∈ C1() is non-negative. We completely analyse the existence and non-existence of (positive) classical solutions in terms of the parameter λ. We show that there exist solutions for every λ when α = 0 and the integral 1/g(s)ds = ∞, or when α > 0 and the integral s/g(s)ds = ∞. Conversely, when the respective integrals converge and h is non-trivial on ∂Ω, existence depends on the size of λ. Moreover, non-existence holds for large λ. Our proofs mainly rely on comparison arguments, and on the construction of suitable supersolutions in annuli. Our results include some cases where the function g is superquadratic and existence still holds without assuming any smallness condition on λ.

Sign in / Sign up

Export Citation Format

Share Document