An Elliptic Problem with a Lower Order Term Having Singular Behaviour

Daniela Giachetti; François Murat

Bollettino dell'Unione Matematica Italiana (2009)

  • Volume: 2, Issue: 2, page 349-370
  • ISSN: 0392-4041

Abstract

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We prove the existence of distributional solutions to an elliptic problem with a lower order term which depends on the solution u in a singular way and on its gradient D ⁢ u with quadratic growth. The prototype of the problem under consideration is { - Δ ⁢ u + λ ⁢ u = ± | D ⁢ u | 2 | u | k + f in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω , where λ > 0 , k > 0 ; f ⁢ ( x ) ∈ L ∞ ⁢ ( Ω ) , f ⁢ ( x ) ≥ 0 (and so u ≥ 0 ). If 0 < k < 1 , we prove the existence of a solution for both the "+" and the "-" signs, while if k ≥ 1 , we prove the existence of a solution for the "+" sign only.

How to cite

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Giachetti, Daniela, and Murat, François. "An Elliptic Problem with a Lower Order Term Having Singular Behaviour." Bollettino dell'Unione Matematica Italiana 2.2 (2009): 349-370. <http://eudml.org/doc/290568>.

@article{Giachetti2009,
abstract = {We prove the existence of distributional solutions to an elliptic problem with a lower order term which depends on the solution $u$ in a singular way and on its gradient $Du$ with quadratic growth. The prototype of the problem under consideration is $$\begin\{cases\} - \Delta u + \lambda u = \pm \frac\{|Du|^\{2\}\}\{|u|^\{k\}\} + f \quad & \text\{in\} \, \Omega, \\ u=0 & \text\{on\} \, \partial \Omega, \end\{cases\}$$ where $\lambda > 0$, $k > 0$; $f(x) \in L^\{\infty\}(\Omega)$, $f(x) \ge 0$ (and so $u \ge 0$). If $0 < k < 1$, we prove the existence of a solution for both the "+" and the "-" signs, while if $k \ge 1$, we prove the existence of a solution for the "+" sign only.},
author = {Giachetti, Daniela, Murat, François},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {6},
number = {2},
pages = {349-370},
publisher = {Unione Matematica Italiana},
title = {An Elliptic Problem with a Lower Order Term Having Singular Behaviour},
url = {http://eudml.org/doc/290568},
volume = {2},
year = {2009},
}

TY - JOUR
AU - Giachetti, Daniela
AU - Murat, François
TI - An Elliptic Problem with a Lower Order Term Having Singular Behaviour
JO - Bollettino dell'Unione Matematica Italiana
DA - 2009/6//
PB - Unione Matematica Italiana
VL - 2
IS - 2
SP - 349
EP - 370
AB - We prove the existence of distributional solutions to an elliptic problem with a lower order term which depends on the solution $u$ in a singular way and on its gradient $Du$ with quadratic growth. The prototype of the problem under consideration is $$\begin{cases} - \Delta u + \lambda u = \pm \frac{|Du|^{2}}{|u|^{k}} + f \quad & \text{in} \, \Omega, \\ u=0 & \text{on} \, \partial \Omega, \end{cases}$$ where $\lambda > 0$, $k > 0$; $f(x) \in L^{\infty}(\Omega)$, $f(x) \ge 0$ (and so $u \ge 0$). If $0 < k < 1$, we prove the existence of a solution for both the "+" and the "-" signs, while if $k \ge 1$, we prove the existence of a solution for the "+" sign only.
LA - eng
UR - http://eudml.org/doc/290568
ER -

References

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