Existence of renormalized solutions for some degenerate and non-coercive elliptic equations

Youssef Akdim; Mohammed Belayachi; Hassane Hjiaj

Mathematica Bohemica (2023)

  • Volume: 148, Issue: 2, page 255-282
  • ISSN: 0862-7959

Abstract

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This paper is devoted to the study of some nonlinear degenerated elliptic equations, whose prototype is given by t 2 - div ( b ( | u | ) | ∇ u | p - 2 ∇ u ) + d ( | u | ) | ∇ u | p = f - div ( c ( x ) | u | α ) in Ω , u = 0 on ∂ Ω , t where Ω is a bounded open set of ℝ N ( N ≥ 2 ) with 1 < p < N and f ∈ L 1 ( Ω ) , under some growth conditions on the function b ( · ) and d ( · ) , where c ( · ) is assumed to be in L N ( p - 1 ) ( Ω ) . We show the existence of renormalized solutions for this non-coercive elliptic equation, also, some regularity results will be concluded.

How to cite

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Akdim, Youssef, Belayachi, Mohammed, and Hjiaj, Hassane. "Existence of renormalized solutions for some degenerate and non-coercive elliptic equations." Mathematica Bohemica 148.2 (2023): 255-282. <http://eudml.org/doc/299526>.

@article{Akdim2023,
abstract = {This paper is devoted to the study of some nonlinear degenerated elliptic equations, whose prototype is given by \[ \begin\{aligned\}t 2&-\{\rm div\}( b(|u|)|\nabla u|^\{p-2\}\nabla u) + d(|u|)|\nabla u|^\{p\} = f - \{\rm div\}(c(x)|u|^\{\alpha \}) &\quad &\mbox\{in\}\ \Omega ,\\ & u = 0 &\quad &\mbox\{on\}\ \partial \Omega , \end\{aligned\}t \] where $\Omega $ is a bounded open set of $\mathbb \{R\}^N$ ($N\ge 2$) with $1<p<N$ and $f \in L^\{1\}(\Omega ),$ under some growth conditions on the function $b(\cdot )$ and $d(\cdot ),$ where $c(\cdot )$ is assumed to be in $L^\{\frac\{N\}\{(p-1)\}\}(\Omega ).$ We show the existence of renormalized solutions for this non-coercive elliptic equation, also, some regularity results will be concluded.},
author = {Akdim, Youssef, Belayachi, Mohammed, Hjiaj, Hassane},
journal = {Mathematica Bohemica},
keywords = {renormalized solution; nonlinear elliptic equation; non-coercive problem},
language = {eng},
number = {2},
pages = {255-282},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Existence of renormalized solutions for some degenerate and non-coercive elliptic equations},
url = {http://eudml.org/doc/299526},
volume = {148},
year = {2023},
}

TY - JOUR
AU - Akdim, Youssef
AU - Belayachi, Mohammed
AU - Hjiaj, Hassane
TI - Existence of renormalized solutions for some degenerate and non-coercive elliptic equations
JO - Mathematica Bohemica
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 148
IS - 2
SP - 255
EP - 282
AB - This paper is devoted to the study of some nonlinear degenerated elliptic equations, whose prototype is given by \[ \begin{aligned}t 2&-{\rm div}( b(|u|)|\nabla u|^{p-2}\nabla u) + d(|u|)|\nabla u|^{p} = f - {\rm div}(c(x)|u|^{\alpha }) &\quad &\mbox{in}\ \Omega ,\\ & u = 0 &\quad &\mbox{on}\ \partial \Omega , \end{aligned}t \] where $\Omega $ is a bounded open set of $\mathbb {R}^N$ ($N\ge 2$) with $1<p<N$ and $f \in L^{1}(\Omega ),$ under some growth conditions on the function $b(\cdot )$ and $d(\cdot ),$ where $c(\cdot )$ is assumed to be in $L^{\frac{N}{(p-1)}}(\Omega ).$ We show the existence of renormalized solutions for this non-coercive elliptic equation, also, some regularity results will be concluded.
LA - eng
KW - renormalized solution; nonlinear elliptic equation; non-coercive problem
UR - http://eudml.org/doc/299526
ER -

References

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  1. Alvino, A., Boccardo, L., Ferone, V., Orsina, L., Trombetti, G., 10.1007/s10231-002-0056-y, Ann. Mat. Pura Appl., IV. Ser. 182 (2003), 53-79. (2003) Zbl1105.35040MR1970464DOI10.1007/s10231-002-0056-y
  2. Alvino, A., Ferone, V., Trombetti, G., A priori estimates for a class of nonuniformly elliptic equations, Atti Semin. Mat. Fis. Univ. Modena 46 (1998), 381-391. (1998) Zbl0911.35025MR1645729
  3. Ali, M. Ben Cheikh, Guibé, O., Nonlinear and non-coercive elliptic problems with integrable data, Adv. Math. Sci. Appl. 16 (2006), 275-297. (2006) Zbl1215.35066MR2253236
  4. Bénilan, P., Boccardo, L., Gallouët, T., Gariepy, R., Pierre, M., Vazquez, J. L., An L 1 -theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 22 (1995), 241-273. (1995) Zbl0866.35037MR1354907
  5. Bensoussan, A., Boccardo, L., Murat, F., 10.1016/S0294-1449(16)30342-0, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 5 (1988), 347-364. (1988) Zbl0696.35042MR0963104DOI10.1016/S0294-1449(16)30342-0
  6. Blanchard, D., Guibé, O., 10.1142/S0219530504000369, Anal. Appl., Singap. 2 (2004), 227-246. (2004) Zbl1129.35370MR2070448DOI10.1142/S0219530504000369
  7. Boccardo, L., Dall'Aglio, A., Orsina, L., Existence and regularity results for some elliptic equations with degenerate coercivity, Atti Semin. Mat. Fis. Univ. Modena 46 (1998), 51-81. (1998) Zbl0911.35049MR1645710
  8. Boccardo, L., Gallouet, T., 10.1016/0362-546X(92)90022-7, Nonlinear Anal., Theory Methods Appl. 19 (1992), 573-579. (1992) Zbl0795.35031MR1183664DOI10.1016/0362-546X(92)90022-7
  9. Croce, G., The regularizing effects of some lower order terms in an elliptic equation with degenerate coercivity, Rend. Mat. Appl., VII. Ser. 27 (2007), 299-314. (2007) Zbl1147.35043MR2398428
  10. Vecchio, T. Del, Posteraro, M. R., Existence and regularity results for nonlinear elliptic equations with measure data, Adv. Differ. Equ. 1 (1996), 899-917. (1996) Zbl0856.35044MR1392010
  11. Pietra, F. Della, Existence results for non-uniformly elliptic equations with general growth in the gradient, Differ. Integral Equ. 21 (2008), 821-836. (2008) Zbl1224.35117MR2483336
  12. Droniou, J., 10.1023/A:1015709329011, Potential Anal. 17 (2002), 181-203. (2002) Zbl1161.35362MR1908676DOI10.1023/A:1015709329011
  13. Droniou, J., 10.1081/PDE-120019377, Commun. Partial Differ. Equations 28 (2003), 129-153. (2003) Zbl1094.35046MR1974452DOI10.1081/PDE-120019377
  14. Guibé, O., Mercaldo, A., 10.1007/s11118-006-9011-7, Potential Anal. 25 (2006), 223-258. (2006) Zbl1198.35072MR2255346DOI10.1007/s11118-006-9011-7
  15. Guibé, O., Mercaldo, A., 10.1090/S0002-9947-07-04139-6, Trans. Am. Math. Soc. 360 (2008), 643-669. (2008) Zbl1156.35042MR2346466DOI10.1090/S0002-9947-07-04139-6
  16. Leone, C., Porretta, A., 10.1016/S0362-546X(96)00323-9, Nonlinear Anal., Theory Methods Appl. 32 (1998), 325-334. (1998) Zbl1155.35352MR1610574DOI10.1016/S0362-546X(96)00323-9
  17. Lions, J. L., Quelques méthodes de résolution des problèmes aux limites non linéaires, Etudes mathematiques. Dunod, Gauthier-Villars, Paris (1969), French. (1969) Zbl0189.40603MR0259693
  18. Maderna, C., Pagani, C. D., Salsa, S., 10.1016/0022-0396(92)90083-Y, J. Differ. Equations 97 (1992), 54-70. (1992) Zbl0785.35039MR1161311DOI10.1016/0022-0396(92)90083-Y
  19. Murat, F., Soluciones renormalizadas de EDP elipticas non lineales, Technical Report R93023, Laboratoire d'Analyse Numérique, Paris (1993), French. (1993) 
  20. Porretta, A., Nonlinear equations with natural growth terms and measure data, Electron. J. Differ. Equ. Conf. 09 (2002), 183-202. (2002) Zbl1109.35341MR1976695
  21. León, S. Segura de, Existence and uniqueness for L 1 data of some elliptic equations with natural growth, Adv. Differ. Equ. 8 (2003), 1377-1408. (2003) Zbl1158.35365MR2016651
  22. Trombetti, C., 10.1023/A:1021884903872, Potential Anal. 18 (2003), 391-404. (2003) Zbl1040.35010MR1953268DOI10.1023/A:1021884903872
  23. Zou, W., 10.1186/s13660-015-0799-9, J. Inequal. Appl. 2015 (2015), Article ID 294, 23 pages. (2015) Zbl1336.35167MR3399257DOI10.1186/s13660-015-0799-9

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