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Existence of renormalized solutions for some degenerate and non-coercive elliptic equations. (English) Zbl 07729577

Summary: This paper is devoted to the study of some nonlinear degenerated elliptic equations, whose prototype is given by \[ \begin{aligned} & -\operatorname{div}(b(|u|)|\nabla u|^{p-2}\nabla u)+d(|u|)|\nabla u|^{p}=f-\operatorname{div}(c(x)|u|^{\alpha}) &\quad &\mbox {in}\ \Omega ,\\ & u=0 &\quad &\mbox {on}\ \partial\Omega, \end{aligned} \] where \(\Omega\) is a bounded open set of \(\mathbb{R}^N\) (\(N\geq 2\)) 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.

MSC:

35J62 Quasilinear elliptic equations
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35J70 Degenerate elliptic equations
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