Uniqueness of renormalized solutions to nonlinear elliptic equations with a lower order term and right-hand side in L1(Ω)

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

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • 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 N , N > 2, 2-1/N < p < N , a belongs to L∞(Ω), a ( x ) α 0 > 0 , f is a function in L1(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 L1(Ω) ." ESAIM: Control, Optimisation and Calculus of Variations 8 (2010): 239-272. <http://eudml.org/doc/90648>.

@article{Betta2010,
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
$$ \left\\{ - \hbox\{div\}( a(x)(1+|\nabla u|^\{2\})^\{\frac\{p-2\}\{2\}\}\nabla u) +b(x)(1+|\nabla u|^\{2\})^\{\frac\{\lambda\}\{2\}\} =f \hbox\{in\} \quad \Omega, u=0 \hbox\{on\} \quad \partial\Omega, \right. $$
where Ω is a bounded open subset of $\{\mathbb\{R\}\}^N$, N > 2, 2-1/N < p < N , a belongs to L∞(Ω), $a(x) \ge \alpha_0>0$, f is a function in L1(Ω), b is a function in $L^r(\Omega)$ and 0 ≤ λ < λ *(N,p,r), for some r and λ *(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 L1.; uniqueness; nonlinear elliptic equations; data in },
language = {eng},
month = {3},
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 L1(Ω) },
url = {http://eudml.org/doc/90648},
volume = {8},
year = {2010},
}

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 L1(Ω)
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
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
$$ \left\{ - \hbox{div}( a(x)(1+|\nabla u|^{2})^{\frac{p-2}{2}}\nabla u) +b(x)(1+|\nabla u|^{2})^{\frac{\lambda}{2}} =f \hbox{in} \quad \Omega, u=0 \hbox{on} \quad \partial\Omega, \right. $$
where Ω is a bounded open subset of ${\mathbb{R}}^N$, N > 2, 2-1/N < p < N , a belongs to L∞(Ω), $a(x) \ge \alpha_0>0$, f is a function in L1(Ω), b is a function in $L^r(\Omega)$ and 0 ≤ λ < λ *(N,p,r), for some r and λ *(N,p,r).
LA - eng
KW - Uniqueness; nonlinear elliptic equations; noncoercive problems; data in L1.; uniqueness; nonlinear elliptic equations; data in
UR - http://eudml.org/doc/90648
ER -

References

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  2. M.F. Betta, A. Mercaldo, F. Murat and M.M. Porzio, Existence and uniqueness results for nonlinear elliptic problems with a lower order term and measure datum. C. R. Acad. Sci. Paris Sér. I Math. 332 (to appear).  Zbl1016.35026
  3. M.F. Betta, A. Mercaldo, F. Murat and M.M. Porzio, Existence of renormalized solutions to nonlinear elliptic equations with a lower order term and right-hand side measure. J. Math. Pures Appl. (to appear).  Zbl1165.35365
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