Elliptic equations with nonlinear absorption depending on the
solution and its gradient

J 2015

Elliptic equations with nonlinear absorption depending on the solution and its gradient

MARCUS, Moshe and Phuoc-Tai NGUYEN

Basic information

Original name

Elliptic equations with nonlinear absorption depending on the solution and its gradient

Authors

MARCUS, Moshe and Phuoc-Tai NGUYEN

Edition

Proceedings of the London Mathematical Society, England, Oxford University Press, 2015, 0024-6115

Other information

Language

English

Type of outcome

Article in a journal

Field of Study

10101 Pure mathematics

Country of publisher

United Kingdom of Great Britain and Northern Ireland

Confidentiality degree

is not subject to a state or trade secret

References:

URL

Impact factor

Impact factor: 1.079

Organization unit

Faculty of Science

DOI

https://doi.org/10.1112/plms/pdv020

UT WoS

000359643300007

Keywords in English

quasilinear equations;boundary singularities;Radon measures;Borel measures;weak singularities;strong singularities;boundary trace;removability

Abstract

In the original language

We study positive solutions of equation (E1) -Delta u + u(p)vertical bar del u vertical bar(q) = 0 (0 <= p, 0 <= q <= 2, p + q > 1) and (E-2) -Delta u + u(p) + vertical bar Delta u vertical bar(q) = 0 (p > 1, 1 < q <= 2) in a smooth bounded domain Omega subset of R-N. We obtain a sharp condition on p and q under which, for every positive, finite Borel measure mu on partial derivative Omega, there exists a solution such that u = mu on partial derivative Omega. Furthermore, if the condition mentioned above fails, then any isolated point singularity on partial derivative Omega is removable, namely, there is no positive solution that vanishes on partial derivative Omega everywhere except at one point. With respect to (E2), we also prove uniqueness and discuss solutions that blow up on a compact subset of partial derivative Omega. In both cases, we obtain a classification of positive solutions with an isolated boundary singularity. Finally, in Appendix A a uniqueness result for a class of quasilinear equations is provided. This class includes (E1) when p = 0 but not the general case.

Cite

MARCUS, Moshe and Phuoc-Tai NGUYEN. Elliptic equations with nonlinear absorption depending on the solution and its gradient. Proceedings of the London Mathematical Society. England: Oxford University Press, 2015, 111/2015, No 1, p. 205-239. ISSN 0024-6115. Available from: https://doi.org/10.1112/plms/pdv020.

@article{1448778,
   author = {Marcus, Moshe and Nguyen, PhuocandTai},
   article_location = {England},
   article_number = {1},
   doi = {https://doi.org/10.1112/plms/pdv020},
   keywords = {quasilinear equations;boundary singularities;Radon measures;Borel measures;weak singularities;strong singularities;boundary trace;removability},
   language = {eng},
   issn = {0024-6115},
   journal = {Proceedings of the London Mathematical Society},
   title = {Elliptic equations with nonlinear absorption depending on the solution and its gradient},
   url = {https://londmathsoc.onlinelibrary.wiley.com/doi/full/10.1112/plms/pdv020},
   volume = {111/2015},
   year = {2015}
}

TY  - JOUR
ID  - 1448778
AU  - Marcus, Moshe - Nguyen, Phuoc-Tai
PY  - 2015
TI  - Elliptic equations with nonlinear absorption depending on the solution and its gradient
JF  - Proceedings of the London Mathematical Society
VL  - 111/2015
IS  - 1
SP  - 205-239
EP  - 205-239
PB  - Oxford University Press
SN  - 00246115
KW  - quasilinear equations;boundary singularities;Radon measures;Borel measures;weak singularities;strong singularities;boundary trace;removability
UR  - https://londmathsoc.onlinelibrary.wiley.com/doi/full/10.1112/plms/pdv020
L2  - https://londmathsoc.onlinelibrary.wiley.com/doi/full/10.1112/plms/pdv020
N2  - We study positive solutions of equation (E1) -Delta u + u(p)vertical bar del u vertical bar(q) = 0 (0 <= p, 0 <= q <= 2, p + q > 1) and (E-2) -Delta u + u(p) + vertical bar Delta u vertical bar(q) = 0 (p > 1, 1 < q <= 2) in a smooth bounded domain Omega subset of R-N. We obtain a sharp condition on p and q under which, for every positive, finite Borel measure mu on partial derivative Omega, there exists a solution such that u = mu on partial derivative Omega. Furthermore, if the condition mentioned above fails, then any isolated point singularity on partial derivative Omega is removable, namely, there is no positive solution that vanishes on partial derivative Omega everywhere except at one point. With respect to (E2), we also prove uniqueness and discuss solutions that blow up on a compact subset of partial derivative Omega. In both cases, we obtain a classification of positive solutions with an isolated boundary singularity. Finally, in Appendix A a uniqueness result for a class of quasilinear equations is provided. This class includes (E1) when p = 0 but not the general case.
ER  -

MARCUS, Moshe and Phuoc-Tai NGUYEN. Elliptic equations with nonlinear absorption depending on the solution and its gradient. \textit{Proceedings of the London Mathematical Society}. England: Oxford University Press, 2015, 111/2015, No~1, p.~205-239. ISSN~0024-6115. Available from: https://doi.org/10.1112/plms/pdv020.

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