scholarly journals Entropy Solutions for Nonlinear Elliptic Anisotropic Homogeneous Neumann Problem

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
B. K. Bonzi ◽  
S. Ouaro ◽  
F. D. Y. Zongo
Keyword(s):  
Weak Solution ◽  
Neumann Problem ◽  
Elliptic Equations ◽  
Existence And Uniqueness ◽  
Entropy Solution ◽  
Boundary Value ◽  
Entropy Solutions ◽  
Approximation Methods ◽  
Nonlinear Elliptic ◽  
Anisotropic Elliptic Equations

We prove the existence and uniqueness of entropy solution for nonlinear anisotropic elliptic equations with Neumann homogeneous boundary value condition for -data. We prove first, by using minimization techniques, the existence and uniqueness of weak solution when the data is bounded, and by approximation methods, we prove a result of existence and uniqueness of entropy solution.

scholarly journals On Entropy Solutions of Anisotropic Elliptic Equations with Variable Nonlinearity Indices

2017 ◽  
Vol 63 (3) ◽  
pp. 475-493 ◽  
Author(s):  
L M Kozhevnikova
Keyword(s):  
Dirichlet Problem ◽  
Sobolev Spaces ◽  
Elliptic Equations ◽  
Existence And Uniqueness ◽  
Unbounded Domains ◽  
Second Order ◽  
Entropy Solutions ◽  
Anisotropic Sobolev Spaces ◽  
Right Hand ◽  
Anisotropic Elliptic Equations

For a certain class of second-order anisotropic elliptic equations with variable nonlinearity indices and L1 right-hand side we consider the Dirichlet problem in arbitrary unbounded domains. We prove the existence and uniqueness of entropy solutions in anisotropic Sobolev spaces with variable indices.

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 Entropy Solutions of Doubly Nonlinear Fractional Laplace Equations

2021 ◽  
Vol 76 (4) ◽  
Author(s):  
Niklas Grossekemper ◽  
Petra Wittbold ◽  
Aleksandra Zimmermann
Keyword(s):  
Elliptic Equations ◽  
Existence And Uniqueness ◽  
Fractional Laplacian ◽  
Nonlinear Elliptic Equations ◽  
Entropy Solutions ◽  
First Order ◽  
Nonlinear Elliptic ◽  
Laplace Equations ◽  
Whole Space ◽  
Doubly Nonlinear

AbstractIn this contribution, we study a class of doubly nonlinear elliptic equations with bounded, merely integrable right-hand side on the whole space $$\mathbb {R}^N$$ R N . The equation is driven by the fractional Laplacian $$(-\varDelta )^{\frac{s}{2}}$$ ( - Δ ) s 2 for $$s\in (0,1]$$ s ∈ ( 0 , 1 ] and a strongly continuous nonlinear perturbation of first order. It is well known that weak solutions are in genreral not unique in this setting. We are able to prove an $$L^1$$ L 1 -contraction and comparison principle and to show existence and uniqueness of entropy solutions.

scholarly journals Entropy and renormalized solutions for some nonlinear anisotropic elliptic equations with variable exponents and L 1-data

2021 ◽  
Vol 7 (2) ◽  
pp. 277-298
Author(s):  
Mostafa El Moumni ◽  
Deval Sidi Mohamed
Keyword(s):  
Elliptic Equations ◽  
Nonlinear Elliptic Equations ◽  
Variable Exponents ◽  
Renormalized Solutions ◽  
Nonlinear Elliptic ◽  
Anisotropic Elliptic Equations

Abstract We prove in this paper some existence and unicity results of entropy and renormalized solutions for some nonlinear elliptic equations with general anisotropic diffusivities and variable exponents. The data are assumed to be merely integrable.

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