# On Some Nonlinear Partial Differential Equations Involving the “1”-Laplacian and Critical Sobolev Exponent

• Volume: 4, page 667-686
• ISSN: 1292-8119

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## Abstract

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Let Ω be a smooth bounded domain in ${𝐑}^{n}$, n > 1, let a and f be continuous functions on $\overline{\Omega }$, ${1}^{☆}=\frac{n}{n-1}$. We are concerned here with the existence of solution in $BV\left(\Omega \right)$, positive or not, to the problem: $\left\{\begin{array}{cccc}\hfill -\mathrm{div}\phantom{\rule{4pt}{0ex}}\sigma +a\left(x\right)sign\phantom{\rule{4pt}{0ex}}u& amp;={f|u|}^{{1}^{☆}-2}u\sigma .\nabla u\hfill & amp;=|\nabla u|\phantom{\rule{4pt}{0ex}}\mathrm{in}\phantom{\rule{4pt}{0ex}}\Omega u\phantom{\rule{4pt}{0ex}}\mathrm{is}\phantom{\rule{4pt}{0ex}}\mathrm{not}\phantom{\rule{4pt}{0ex}}\mathrm{identically}\phantom{\rule{4pt}{0ex}}\mathrm{zero},& amp;-\sigma .n\left(u\right)=|u|\phantom{\rule{4pt}{0ex}}\mathrm{on}\phantom{\rule{4pt}{0ex}}\partial \Omega .\end{array}\right\$ This problem is closely related to the extremal functions for the problem of the best constant of ${W}^{1,1}\left(\Omega \right)$ into ${L}^{\frac{N}{N-1}}\left(\Omega \right)$.

## How to cite

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Demengel, Françoise. "On Some Nonlinear Partial Differential Equations Involving the “1”-Laplacian and Critical Sobolev Exponent." ESAIM: Control, Optimisation and Calculus of Variations 4 (2010): 667-686. <http://eudml.org/doc/197312>.

@article{Demengel2010,
abstract = { Let Ω be a smooth bounded domain in $\{\bf R\}^n$, n > 1, let a and f be continuous functions on $\bar\Omega$, $1^\star = \{n\over n-1\}$. We are concerned here with the existence of solution in $BV(\Omega)$, positive or not, to the problem: $$\left\\{ \begin\{array\}\{rl\} -\{\rm div\}\ \sigma+a(x) sign\ u &amp;= f|u|^\{1^\star-2\} u\cr \sigma.\nabla u &amp;= |\nabla u|\ \{\rm in\}\ \Omega\cr u\ \{\rm is\ not \ identically\ zero\}, &amp;-\sigma.n (u) = |u|\ \{\rm on \}\ \partial \Omega.\end\{array\}\right.$$ This problem is closely related to the extremal functions for the problem of the best constant of $W^\{1,1\}(\Omega)$ into $L^\{N\over N-1\}(\Omega)$. },
author = {Demengel, Françoise},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {BV functions; best constant for Sobolev embeddings.; 1-Laplacian; critical Sobolev exponent; existence},
language = {eng},
month = {3},
pages = {667-686},
publisher = {EDP Sciences},
title = {On Some Nonlinear Partial Differential Equations Involving the “1”-Laplacian and Critical Sobolev Exponent},
url = {http://eudml.org/doc/197312},
volume = {4},
year = {2010},
}

TY - JOUR
AU - Demengel, Françoise
TI - On Some Nonlinear Partial Differential Equations Involving the “1”-Laplacian and Critical Sobolev Exponent
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 4
SP - 667
EP - 686
AB - Let Ω be a smooth bounded domain in ${\bf R}^n$, n > 1, let a and f be continuous functions on $\bar\Omega$, $1^\star = {n\over n-1}$. We are concerned here with the existence of solution in $BV(\Omega)$, positive or not, to the problem: $$\left\{ \begin{array}{rl} -{\rm div}\ \sigma+a(x) sign\ u &amp;= f|u|^{1^\star-2} u\cr \sigma.\nabla u &amp;= |\nabla u|\ {\rm in}\ \Omega\cr u\ {\rm is\ not \ identically\ zero}, &amp;-\sigma.n (u) = |u|\ {\rm on }\ \partial \Omega.\end{array}\right.$$ This problem is closely related to the extremal functions for the problem of the best constant of $W^{1,1}(\Omega)$ into $L^{N\over N-1}(\Omega)$.
LA - eng
KW - BV functions; best constant for Sobolev embeddings.; 1-Laplacian; critical Sobolev exponent; existence
UR - http://eudml.org/doc/197312
ER -

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