Properties of the 1-polyharmonic operator in the whole space and applications to nonlinear elliptic equations

2021 ◽  
pp. 125843
Author(s):  
Sami Aouaoui ◽  
Mariem Dhifet
Keyword(s):  
Elliptic Equations ◽  
Nonlinear Elliptic Equations ◽  
Polyharmonic Operator ◽  
Nonlinear Elliptic ◽  
Whole Space

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.

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

An asymptotic treatment for non-convex fully nonlinear elliptic equations: Global Sobolev and BMO type estimates

2019 ◽  
Vol 21 (07) ◽  
pp. 1850053 ◽  
Author(s):  
J. V. da Silva ◽  
G. C. Ricarte
Keyword(s):  
Elliptic Equations ◽  
A Priori Estimates ◽  
Integrability Condition ◽  
A Priori ◽  
Nonlinear Elliptic Equations ◽  
Borderline Case ◽  
Fully Nonlinear Elliptic Equations ◽  
Fully Nonlinear ◽  
Text Type ◽  
Nonlinear Elliptic

In this paper, we establish global Sobolev a priori estimates for [Formula: see text]-viscosity solutions of fully nonlinear elliptic equations as follows: [Formula: see text] by considering minimal integrability condition on the data, i.e. [Formula: see text] for [Formula: see text] and a regular domain [Formula: see text], and relaxed structural assumptions (weaker than convexity) on the governing operator. Our approach makes use of techniques from geometric tangential analysis, which consists in transporting “fine” regularity estimates from a limiting operator, the Recession profile, associated to [Formula: see text] to the original operator via compactness methods. We devote special attention to the borderline case, i.e. when [Formula: see text]. In such a scenery, we show that solutions admit [Formula: see text] type estimates for their second derivatives.

MULTIPLE SOLUTIONS FOR NONLINEAR ELLIPTIC EQUATIONS ON COMPACT RIEMANNIAN MANIFOLDS

2007 ◽  
Vol 18 (09) ◽  
pp. 1071-1111 ◽  
Author(s):  
JÉRÔME VÉTOIS
Keyword(s):  
Continuous Function ◽  
Riemannian Manifolds ◽  
Elliptic Equations ◽  
Multiple Solutions ◽  
Nonlinear Elliptic Equations ◽  
Beltrami Operator ◽  
Hölder Continuous ◽  
Nonlinear Elliptic

Let (M,g) be a smooth compact Riemannian n-manifold, n ≥ 4, and h be a Holdër continuous function on M. We prove multiplicity of changing sign solutions for equations like Δg u + hu = |u|2* - 2 u, where Δg is the Laplace–Beltrami operator and 2* = 2n/(n - 2) is critical from the Sobolev viewpoint.

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