Abstract：In this paper, we consider the following nonlinear elliptic equation with degenerate coercivity and lower order term in the setting of the weighted Sobolev space. We investigate the existence of the renormalized solutions in W01,p(Ω,ω) by the truncation method. With the help of Marcinkiewicz estimate, through some priori estimates for the sequence of solutions of the approximate problem, we prove that un converges in measure. Then we choose suitable test functions for the approximate equation and obtain the needed estimates. Finally, through a limit process, we obtain the existence of renormalized solutions to problem.
代丽丽. 一类具退化强制的椭圆方程在加权Sobolev空间中重整化解的存在性[J]. 浙江大学学报（理学版）, 2018, 45(6): 673-678.
DAI Lili. Existence of renormalized solutions for a degenerate elliptic equation with degenerate coercivity in weighted Sobolev spaces. Journal of ZheJIang University(Science Edition), 2018, 45(6): 673-678.
[1] BOCCARDO L, DALL'AGLIO A, ORSINA L. Existence and regularity results for some elliptic equations with degenerate coercivity[J].Atti del Semimario Matematico e Fisico dell' Universitã di Modena, 1998, 46:51-82.
[2] DAIL L, GAO W J, LI Z Q. Existence of solutions for degenerate elliptic problems in weighted Sobolev space[J]. Journal of Function Spaces, 2015, 2015(71):1-9.
[3] GIACHETTI D, PORZIO M M. Existence results for some nonuniformly elliptic equations with irregular data[J]. Journal of Mathematical Analysis and Applications, 2001, 257(1):100-130.
[4] GIACHETTI D, PORZIO M M. Elliptic equations with degenerate coercivity:Gradient regularity[J].Acta Mathematica Sinica, 2003, 19(2):349-370.
[5] PORRETTA A. Uniqueness and homogenization for a class of non coercive operators in divergence form[J].Atti del Semimario Matematico e Fisico dell' Universitã di Modena, 1998, 46:915-936.
[6] CROCE G. The regularizing effects of some lower order terms in an elliptic equation with degenerate coercivity[J]. Rendiconti di Matematica, 2007, 27:299-314.
[7] LEONE C, PORRETTA A. Entropy solutions for nonlinear elliptic equations in L1(Ω)[J]. Nonlinear Analysis Theory Methods and Applications, 1998, 32(3):325-334.
[8] BOCCARDO L, CROCE G, ORSINA L. Nonlinear degenerate elliptic problems with W01,1(Ω) solutions[J]. Manuscripta Mathematica, 2012, 137(3/4):419-439.
[9] GOL'DSHTEIN V, UKHLOV A. Weighted Sobolev spaces and embedding theorems[J].Transactions of the American Mathematical Society, 2009, 361(7):3829-3850.
[10] DRÁBEK P, KUFNER A, MUSTONEN V. Pseudo-monotonicity and degenerated or singular elliptic operators[J]. Bulletin of the Australian Mathematical Society,1998,58(2):213-221.
[11] AZORERO J P G, ALONSO I P. Hardy inequalities and some critical elliptic and parabolic problems[J]. Journal of Differential Equations, 1998, 144(2):441-476.
[12] BLANCHARD D. Truncations and monotonicity methods for parabolic equations[J].Nonlinear Analysis:Theory, Methods and Applications, 1993, 21(10):725-743.
[13] BLANCHARD D,MURAT F, REDWANE H. Existence and uniqueness of a renormalized solution for a fairly general class of nonlinear parabolic problems[J]. Journal of Differential Equations, 2001, 177(2):331-374.
[14] BENILAN P, BREZIS H, CRANDALL M G. A semilinear equation in L1(RN)[J]. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze(Serie IV), 1975, 2(4):523-555.
[15] LIONS J L. Quelques Méthodes de Résolution des Problemes aux Limites Nonlinéaires[M]. Paris:Dunod, 1969.
[16] SCHAFT A V D. Nonlinear systems analysis:M. Vidyasagar[J]. Automatica, 1994, 30(10):1631-1632.
2018, Vol. 45 
