Singular nonlinear elliptic equations inRn

This paper deals with existence, uniqueness and regularity of positive generalized solutions of singular nonlinear equations of the form−Δu+a(x)u=h(x)u−γinRnwherea,hare given, not necessarily continuous functions, andγis a positive number. We explore both situations wherea,hare radial functions, withabeing eventually identically zero, and cases where no symmetry is required from eitheraorh. Schauder's fixed point theorem, combined with penalty arguments, is exploited.

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Borderline regularity for fully nonlinear equations in Dini domains

Advances in Calculus of Variations ◽

10.1515/acv-2020-0030 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Karthik Adimurthi ◽

Agnid Banerjee

Keyword(s):

Nonlinear Equations ◽

Elliptic Equations ◽

Nonlinear Elliptic Equations ◽

Fully Nonlinear Elliptic Equations ◽

Fully Nonlinear Equations ◽

Fully Nonlinear ◽

Nonlinear Elliptic ◽

Regularity Estimates ◽

Compactness Arguments ◽

Gradient Regularity

AbstractIn this paper, we prove borderline gradient continuity of viscosity solutions to fully nonlinear elliptic equations at the boundary of a C^{1,\mathrm{Dini}}-domain. Our main result constitutes the boundary analogue of the borderline interior gradient regularity estimates established by P. Daskalopoulos, T. Kuusi and G. Mingione. We however mention that, differently from the approach used there which is based on W^{1,q} estimates, our proof is slightly more geometric and is based on compactness arguments inspired by the techniques in the fundamental works of Luis Caffarelli.

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On the boundedness of generalized solutions of higher-order nonlinear elliptic equations with data from an Orlicz–Zygmund class

Mathematical Notes ◽

10.1134/s0001434616050229 ◽

2016 ◽

Vol 99 (5-6) ◽

pp. 840-850 ◽

Cited By ~ 2

Author(s):

M. V. Voitovich

Keyword(s):

Elliptic Equations ◽

Nonlinear Elliptic Equations ◽

Higher Order ◽

Generalized Solutions ◽

Zygmund Class ◽

Nonlinear Elliptic

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On Generalized and Viscosity Solutions of Nonlinear Elliptic Equations

Advanced Nonlinear Studies ◽

10.1515/ans-2004-0304 ◽

2004 ◽

Vol 4 (3) ◽

Author(s):

David Hartenstine ◽

Klaus Schmitt

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Elliptic Equations ◽

Nonlinear Elliptic Equations ◽

Elliptic Partial Differential Equations ◽

Generalized Solutions ◽

Differential Inequalities ◽

Nonlinear Elliptic ◽

Relationship Of ◽

The Relationship

AbstractThere are many notions of solutions of nonlinear elliptic partial differential equations. This paper is concerned with solutions which are obtained as suprema (or infima) of so-called subfunctions (superfunctions) or viscosity subsolutions (viscosity supersolutions). The paper also explores the relationship of these (generalized) solutions of differential inequalities and provides a relevant example for which existence questions have been studied using these concepts.

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Nonnegative Solutions for Weakly Nonlinear Elliptic Equations

Canadian Journal of Mathematics ◽

10.4153/cjm-1984-006-4 ◽

1984 ◽

Vol 36 (1) ◽

pp. 71-83 ◽

Cited By ~ 1

Author(s):

Walter Allegretto

Keyword(s):

Elliptic Equations ◽

Bounded Solution ◽

Smooth Boundary ◽

Nonlinear Elliptic Equations ◽

Generalized Solutions ◽

Main Criterion ◽

Weakly Nonlinear ◽

Nonlinear Elliptic ◽

Nonnegative Solutions ◽

Image Position

Let x = (x1, … xn) denote a point of Euclidean n space En and set Di = ∂/∂xi for i = 1, … n. Let Ω denote an exterior domain in En with smooth boundary and consider in Ω the formal elliptic problem:1We first consider the problem of finding nonnegative generalized solutions of (1) when τ ≧ 0, τ ≢ 0, and r(x) ≡ 0. Under more stringent conditions on the coefficients and for suitable r(x), we then show the existence of a locally bounded solution. Next, we show that, under stronger assumptions, our main criterion is also necessary. The final arguments are devoted to the consideration of illustrative examples.

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Applications of Sub-Supersolution Theorems to Singular Nonlinear Elliptic Problems

Advanced Nonlinear Studies ◽

10.1515/ans-2011-0302 ◽

2011 ◽

Vol 11 (3) ◽

Cited By ~ 3

Author(s):

Nguyen Hoang Loc ◽

Klaus Schmitt

Keyword(s):

Boundary Value Problems ◽

Nonlinear Equations ◽

Elliptic Equations ◽

Semilinear Elliptic Equations ◽

Singular Boundary ◽

Singular Boundary Value Problems ◽

Nonlinear Elliptic Problems ◽

Nonlinear Elliptic ◽

Semilinear Elliptic ◽

Singular Nonlinear Equations

AbstractThe results presented here were motivated by several recent papers on singular boundary value problems for semilinear elliptic equations with convection terms. We present extensions which cover singular nonlinear equations (mainly equations involving the p−Laplacian) containing convection terms. The results obtained are proved using sub- and supersolution theorems (motivated by the results in [18, 19, 20, 23]) and the construction of a well-ordered pair of such using a principal eigenfunction of the p−Laplacian.

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