Uniqueness of renormalized solutions to nonlinear elliptic equations with a lower order term and right-hand side in L 1 ( Ω )

M. F. Betta; A. Mercaldo; F. Murat; M. M. Porzio

ESAIM: Control, Optimisation and Calculus of Variations (2002)

  • Volume: 8, page 239-272
  • ISSN: 1292-8119

Abstract

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In this paper we prove uniqueness results for the renormalized solution, if it exists, of a class of non coercive nonlinear problems whose prototype is - div ( a ( x ) ( 1 + | ∇ u | 2 ) p - 2 2 ∇ u ) + b ( x ) ( 1 + | ∇ u | 2 ) λ 2 = f in Ω , u = 0 on ∂ Ω , where Ω is a bounded open subset of ℝ N , N ≥ 2 , 2 - 1 / N < p < N , a belongs to L ∞ ( Ω ) , a ( x ) ≥ α 0 > 0 , f is a function in L 1 ( Ω ) , b is a function in L r ( Ω ) and 0 ≤ λ < λ * ( N , p , r ) , for some r and λ * ( N , p , r ) .

How to cite

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Betta, M. F., et al. "Uniqueness of renormalized solutions to nonlinear elliptic equations with a lower order term and right-hand side in $L^1(\Omega )$." ESAIM: Control, Optimisation and Calculus of Variations 8 (2002): 239-272. <http://eudml.org/doc/245921>.

@article{Betta2002,
abstract = {In this paper we prove uniqueness results for the renormalized solution, if it exists, of a class of non coercive nonlinear problems whose prototype is\[ \{\left\lbrace \begin\{array\}\{ll\} - \operatorname\{div\}( a(x)(1+|\nabla u|^\{2\})^\{\frac\{p-2\}\{2\}\}\nabla u) +b(x)(1+|\nabla u|^\{2\})^\{\frac\{\lambda \}\{2\}\} =f &\text\{in\} \Omega ,\\ u=0 &\text\{on\} \partial \Omega , \end\{array\}\right.\} \]where $\Omega $ is a bounded open subset of $\{\mathbb \{R\}\}^N$, $N\ge 2$, $2-1/N&lt; p&lt; N$, $a$ belongs to $L^\{\infty \}(\Omega )$, $a(x) \ge \alpha _0&gt;0$, $f$ is a function in $L^1(\Omega ) $, $b$ is a function in $L^r(\Omega )$ and $0\le \lambda &lt;\lambda ^*(N,p,r),$ for some $r$ and $\lambda ^*(N,p,r)$.},
author = {Betta, M. F., Mercaldo, A., Murat, F., Porzio, M. M.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {uniqueness; nonlinear elliptic equations; noncoercive problems; data in $L^1$; data in },
language = {eng},
pages = {239-272},
publisher = {EDP-Sciences},
title = {Uniqueness of renormalized solutions to nonlinear elliptic equations with a lower order term and right-hand side in $L^1(\Omega )$},
url = {http://eudml.org/doc/245921},
volume = {8},
year = {2002},
}

TY - JOUR
AU - Betta, M. F.
AU - Mercaldo, A.
AU - Murat, F.
AU - Porzio, M. M.
TI - Uniqueness of renormalized solutions to nonlinear elliptic equations with a lower order term and right-hand side in $L^1(\Omega )$
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2002
PB - EDP-Sciences
VL - 8
SP - 239
EP - 272
AB - In this paper we prove uniqueness results for the renormalized solution, if it exists, of a class of non coercive nonlinear problems whose prototype is\[ {\left\lbrace \begin{array}{ll} - \operatorname{div}( a(x)(1+|\nabla u|^{2})^{\frac{p-2}{2}}\nabla u) +b(x)(1+|\nabla u|^{2})^{\frac{\lambda }{2}} =f &\text{in} \Omega ,\\ u=0 &\text{on} \partial \Omega , \end{array}\right.} \]where $\Omega $ is a bounded open subset of ${\mathbb {R}}^N$, $N\ge 2$, $2-1/N&lt; p&lt; N$, $a$ belongs to $L^{\infty }(\Omega )$, $a(x) \ge \alpha _0&gt;0$, $f$ is a function in $L^1(\Omega ) $, $b$ is a function in $L^r(\Omega )$ and $0\le \lambda &lt;\lambda ^*(N,p,r),$ for some $r$ and $\lambda ^*(N,p,r)$.
LA - eng
KW - uniqueness; nonlinear elliptic equations; noncoercive problems; data in $L^1$; data in
UR - http://eudml.org/doc/245921
ER -

References

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