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

Boumediene Abdellaoui; Ireneo Peral

Journal of the European Mathematical Society (2006)

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

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topAbdellaoui, 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 -

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