# A note on a critical problem with natural growth in the gradient

• Volume: 008, Issue: 2, page 157-170
• ISSN: 1435-9855

top

## Abstract

top
The paper analyzes the influence on the meaning of natural growth in the gradient of a perturbation by a Hardy potential in some elliptic equations. Indeed, in the case of the Laplacian the natural problem becomes $-\Delta u-{\Lambda }_{N}\frac{u}{{|x|}^{2}}={\left|\nabla u+\frac{N-2}{2}\frac{u}{{|x|}^{2}}x\right|}^{2}{|x|}^{\left(N-2\right)/2}+\lambda f\left(x\right)$ in $\Omega$, $u=0$ on $\partial \Omega$, ${\Lambda }_{N}={\left(\left(N-2\right)/2\right)}^{2}$. This problem is a particular case of problem (2). Notice that $\left(N-2\right)/2$ is optimal as coefficient and exponent on the right hand side.

## How to cite

top

Abdellaoui, Boumediene, and Peral, Ireneo. "A note on a critical problem with natural growth in the gradient." Journal of the European Mathematical Society 008.2 (2006): 157-170. <http://eudml.org/doc/277788>.

@article{Abdellaoui2006,
abstract = {The paper analyzes the influence on the meaning of natural growth in the gradient of a perturbation by a Hardy potential in some elliptic equations. Indeed, in the case of the Laplacian the natural problem becomes $−\Delta u−\Lambda _N\frac\{u\}\{|x|^2\}=\left|\nabla u+\frac\{N−2\}\{2\}\frac\{u\}\{|x|^2\}x\right|^2|x|^\{(N−2)/2\}+\lambda f(x)$ in $\Omega$, $u=0$ on $\partial \Omega$, $\Lambda _N=((N−2)/2)^2$. This problem is a particular case of problem (2). Notice that $(N−2)/2$ is optimal as coefficient and exponent on the right hand side.},
author = {Abdellaoui, Boumediene, Peral, Ireneo},
journal = {Journal of the European Mathematical Society},
keywords = {elliptic equations; Hardy potential; quadratic growth in the gradient; optimal summability; elliptic equations; Hardy potential; quadratic growth in the gradient; optimal summability},
language = {eng},
number = {2},
pages = {157-170},
publisher = {European Mathematical Society Publishing House},
title = {A note on a critical problem with natural growth in the gradient},
url = {http://eudml.org/doc/277788},
volume = {008},
year = {2006},
}

TY - JOUR
AU - Abdellaoui, Boumediene
AU - Peral, Ireneo
TI - A note on a critical problem with natural growth in the gradient
JO - Journal of the European Mathematical Society
PY - 2006
PB - European Mathematical Society Publishing House
VL - 008
IS - 2
SP - 157
EP - 170
AB - The paper analyzes the influence on the meaning of natural growth in the gradient of a perturbation by a Hardy potential in some elliptic equations. Indeed, in the case of the Laplacian the natural problem becomes $−\Delta u−\Lambda _N\frac{u}{|x|^2}=\left|\nabla u+\frac{N−2}{2}\frac{u}{|x|^2}x\right|^2|x|^{(N−2)/2}+\lambda f(x)$ in $\Omega$, $u=0$ on $\partial \Omega$, $\Lambda _N=((N−2)/2)^2$. This problem is a particular case of problem (2). Notice that $(N−2)/2$ is optimal as coefficient and exponent on the right hand side.
LA - eng
KW - elliptic equations; Hardy potential; quadratic growth in the gradient; optimal summability; elliptic equations; Hardy potential; quadratic growth in the gradient; optimal summability
UR - http://eudml.org/doc/277788
ER -

top

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.