The exterior Dirichlet problem for quasi-linear elliptic equations with small boundary data

1987 ◽  
Vol 59 (1) ◽  
pp. 53-62
Author(s):  
Chi -ping Lau
Keyword(s):  
Dirichlet Problem ◽  
Elliptic Equations ◽  
Boundary Data

Gradient estimates for semilinear elliptic equations

1985 ◽  
Vol 100 (1-2) ◽  
pp. 11-17
Author(s):  
Gary M. Lieberman
Keyword(s):  
Dirichlet Problem ◽  
Elliptic Equation ◽  
Elliptic Equations ◽  
Boundary Data ◽  
Smooth Boundary ◽  
Semilinear Elliptic Equation ◽  
Semilinear Elliptic Equations ◽  
Gradient Estimates ◽  
Quadratic Growth ◽  
Semilinear Elliptic

SynopsisEstimates on the gradient of solutions to the Dirichlet problem for a semilinear elliptic equation are given when the nonlinearity in the equation is quadratic with respect to the gradient of the solution. These estimates extend results of F. Tomi to less smooth boundary data and results of the author to the full quadratic growth.

scholarly journals Measure Data Problems for a Class of Elliptic Equations with Mixed Absorption-Reaction

2021 ◽  
Vol 21 (2) ◽  
pp. 261-280
Author(s):  
Marie-Françoise Bidaut-Véron ◽  
Marta Garcia-Huidobro ◽  
Laurent Véron
Keyword(s):  
Dirichlet Problem ◽  
Compact Subset ◽  
Elliptic Equations ◽  
Radon Measure ◽  
Measure Data ◽  
Nonnegative Solutions

Abstract In the present paper, we study the existence of nonnegative solutions to the Dirichlet problem ℒ p , q M ⁢ u := - Δ ⁢ u + u p - M ⁢ | ∇ ⁡ u | q = μ {{\mathcal{L}}^{{M}}_{p,q}u:=-\Delta u+u^{p}-M|\nabla u|^{q}=\mu} in a domain Ω ⊂ ℝ N {\Omega\subset\mathbb{R}^{N}} where μ is a nonnegative Radon measure, when p > 1 {p>1} , q > 1 {q>1} and M ≥ 0 {M\geq 0} . We also give conditions under which nonnegative solutions of ℒ p , q M ⁢ u = 0 {{\mathcal{L}}^{{M}}_{p,q}u=0} in Ω ∖ K {\Omega\setminus K} , where K is a compact subset of Ω, can be extended as a solution of the same equation in Ω.

On the Solution of the Dirichlet Problem with Rational Holomorphic Boundary Data

2006 ◽  
Vol 5 (2) ◽  
pp. 445-457 ◽  
Author(s):  
Peter Ebenfelt ◽  
Michael Viscardi
Keyword(s):  
Dirichlet Problem ◽  
Boundary Data

Multiple solutions for semi-linear corner degenerate elliptic equations with singular potential term

2016 ◽  
Vol 19 (04) ◽  
pp. 1650043 ◽  
Author(s):  
Hua Chen ◽  
Shuying Tian ◽  
Yawei Wei
Keyword(s):  
Dirichlet Problem ◽  
Variational Method ◽  
Elliptic Equations ◽  
Multiple Solutions ◽  
Dirichlet Boundary ◽  
Singular Potential ◽  
Degenerate Elliptic Equations ◽  
Dirichlet Boundary Value Problem ◽  
Potential Term ◽  
Degenerate Elliptic

The present paper is concern with the Dirichlet problem for semi-linear corner degenerate elliptic equations with singular potential term. We first give the preliminary of the framework and then discuss the weighted corner type Hardy inequality. By using the variational method, we prove the existence of multiple solutions for the Dirichlet boundary-value problem.

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