Moderate solutions of semilinear elliptic equations with Hardy potential

MARCUS, Moshe and Phuoc-Tai NGUYEN. Moderate solutions of semilinear elliptic equations with Hardy potential. Annales de l'Institut Henri Poincaré. Analyse Non Linéaire. The Netherlands: Elsevier, 2015, 34/2017, No 1, p. 69-88. ISSN 0294-1449. Available from: https://dx.doi.org/10.1016/j.anihpc.2015.10.001.

Basic information

Original name

Moderate solutions of semilinear elliptic equations with Hardy potential

Authors

MARCUS, Moshe and Phuoc-Tai NGUYEN

Edition

Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, The Netherlands, Elsevier, 2015, 0294-1449

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Other information

Language

English

Type of outcome

Article in a journal

Field of Study

10101 Pure mathematics

Country of publisher

Netherlands

Confidentiality degree

is not subject to a state or trade secret

References:

URL

Impact factor

Impact factor: 2.066

Organization unit

Faculty of Science

DOI

http://dx.doi.org/10.1016/j.anihpc.2015.10.001

UT WoS

000392792800004

Keywords in English

Hardy potential;Martin kernel;Moderate solutions;Normalized boundary trace;Critical exponent;Removable singularities

Tags

International impact, Reviewed

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Abstract

In the original language

Let Omega be a bounded smooth domain in R-N. We study positive solutions of equation (E) - L(mu)u + u(q) = 0 in Omega where L-mu = Delta + mu/delta(2), 0 < mu, q > 1 and delta(x) = dist (x, partial derivative Omega). A positive solution of (E) is moderate if it is dominated by an L-mu-harmonic function. If mu < C-H (Omega) (the Hardy constant for Omega) every positive L-mu-harmonic function can be represented in terms of a finite measure on partial derivative Omega via the Martin representation theorem. However the classical measure boundary trace of any such solution is zero. We introduce a notion of normalized boundary trace by which we obtain a complete classification of the positive moderate solutions of (E) in the subcritical case, 1 < q < q(mu,c). (The critical value depends only on N and mu) For q >= q(mu,c) there exists no moderate solution with an isolated singularity on the boundary. The normalized boundary trace and associated boundary value problems are also discussed in detail for the linear operator L-mu. These results form the basis for the study of the nonlinear problem.