scholarly journals A priori regularity for weak solutions of some nonlinear elliptic equations

1994 ◽  
Vol 11 (6) ◽  
pp. 693-703
Author(s):  
Frank Pacard
Keyword(s):  
Weak Solutions ◽  
Elliptic Equations ◽  
A Priori ◽  
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 boundary blow-up solutions for nonlinear elliptic equations

2003 ◽  
Vol 133 (2) ◽  
pp. 225-235 ◽  
Author(s):  
Amandine Aftalion ◽  
Manuel del Pino ◽  
René Letelier
Keyword(s):  
Elliptic Equations ◽  
Blow Up ◽  
A Priori ◽  
Nonlinear Elliptic Equations ◽  
Positive Parameter ◽  
Number Of Solutions ◽  
Bounded Smooth Domain ◽  
Nonlinear Elliptic ◽  
Bounded Smooth

We consider the problem Δu = λf(u) in Ω, u(x) tends to +∞ as x approaches ∂Ω. Here, Ω is a bounded smooth domain in RN, N ≥ 1 and λ is a positive parameter. In this paper, we are interested in analysing the role of the sign changes of the function f in the number of solutions of this problem. As a consequence of our main result, we find that if Ω is star-shaped and f behaves like f(u) = u(u−a)(u−1) with ½ < a < 1, then there is a solution bigger than 1 for all λ and there exists λ0 > 0 such that, for λ < λ0, there is no positive solution that crosses 1 and, for λ > λ0, at least two solutions that cross 1. The proof is based on a priori estimates, the construction of barriers and topological-degree arguments.

On the viscosity solutions of eigenvalue problems for a class of nonlinear elliptic equations

2019 ◽  
Vol 12 (4) ◽  
pp. 393-421
Author(s):  
Tilak Bhattacharya ◽  
Leonardo Marazzi
Keyword(s):  
Viscosity Solutions ◽  
Elliptic Equations ◽  
Eigenvalue Problems ◽  
A Priori ◽  
Nonlinear Elliptic Equations ◽  
First Eigenvalue ◽  
Nonlinear Elliptic ◽  
Degenerate Elliptic ◽  
First Eigenfunction ◽  
Nonlinear Degenerate

AbstractWe consider viscosity solutions of a class of nonlinear degenerate elliptic equations, involving a parameter, on bounded domains. These arise in the study of eigenvalue problems. We prove comparison principles and a priori supremum bounds for the solutions. We also address the eigenvalue problem and, in many instances, show the existence of the first eigenvalue and an associated positive first eigenfunction.

Elliptic equations on manifolds and isoperimetric inequalities

1990 ◽  
Vol 114 (3-4) ◽  
pp. 213-227 ◽  
Author(s):  
Andrea Cianchi
Keyword(s):  
Riemannian Manifolds ◽  
Elliptic Equations ◽  
A Priori ◽  
Neumann Boundary ◽  
Nonlinear Elliptic Equations ◽  
Isoperimetric Inequalities ◽  
Divergence Form ◽  
Sharp Estimates ◽  
Nonlinear Elliptic ◽  
Homogeneous Neumann Boundary Condition

SynopsisWe consider linear and nonlinear elliptic equations in divergence form on Riemannian manifolds with or without boundary. In the former case we impose a homogeneous Neumann boundary condition. By making use of isoperimetric inequalities for manifolds, we obtain a priori sharp estimates for the decreasing rearrangement of the solutions to such equations. These estimates enable us to derive bounds for suitable norms of the solutions and of their gradients.

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.

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