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Díaz Díaz, Jesús Ildefonso Boccardo, L. Giachetti, D. Murat, F. 2023-06-20T16:56:46Z 2023-06-20T16:56:46Z 1993-11 0022-0396 10.1006/jdeq.1993.1106 https://hdl.handle.net/20.500.14352/57484 http://www.sciencedirect.com/science/article/pii/S002203968371106X http://www.sciencedirect.com/science/journal/00220396/106/ The authors study the nonlinear elliptic equation (*) −div(a(x,u,Du))−div(Φ(u))+g(x,u)=f(x)in Ω with the boundary condition (∗∗) u=0 on ∂Ω, where Ω is a bounded open subset of RN, A(u)=−div(a(x,u,Du)) is a nonlinear operator of Leray-Lions type from W1,p0(Ω) into its dual, Φ∈(C0(R))N, g(x,t)t≥0, and f∈W−1,p′(Ω). No growth hypothesis is assumed on the vector-valued function Φ(u). The term div(Φ(u)) may be meaningless for usual weak solutions, even as a distribution. The authors deal with a weaker form of (∗),(∗∗), solutions of which in W1,p0(Ω) are called the "renormalized solutions'' of the original problem (∗),(∗∗). This weaker form can be formally obtained from (∗) by means of pointwise multiplication by h(u), where h is a C1 function with compact support. They prove the existence of renormalized solutions of (∗),(∗∗) and give sufficient conditions under which a renormalized solution is a usual weak solution. Furthermore, the authors study Ls- and L∞-regularity of renormalized solutions. eng restricted access Existence and regularity of renormalized solutions for some elliptic problems involving derivatives of nonlinear terms journal article