Porretta, A. Existence for elliptic equations in \(L^1\) having lower order terms with natural growth. (English) Zbl 0963.35068 Port. Math. 57, No. 2, 179-190 (2000). Summary: We deal with the following type of nonlinear elliptic equations in a bounded subset \(\Omega\subset \mathbb{R}^N\): \[ \begin{cases} -\text{div}(a(x, u,\nabla u))+ g(x,u,\nabla u)= \chi\quad &\text{in }\Omega,\\ u= 0\quad &\text{on }\partial\Omega,\end{cases}\tag{P} \] where both \(a(x,s,\xi)\) and \(g(x,s,\xi)\) are Carathéodory functions such that \(a(x,s,\cdot)\) is coercive, monotone and has a linear growth, while \(g(x,s,\xi)\) has a quadratic growth with respect to \(\xi\) and satisfies a sign condition on \(s\), that is \(g(x,s,\xi) s\geq 0\) for every \(s\) in \(\mathbb{R}\). The datum \(\chi\) is assumed in \(L^1(\Omega)+ H^{-1}(\Omega)\). We prove the existence of a weak solution \(u\) of (P) which belongs to the Sobolev space \(W^{1,q}_0(\Omega)\) for every \(q< {N\over N-1}\), by adapting to the framework of \(L^1\) data a technique used in [L. Boccardo, F. Murat and J. P. Puel, Existence de solutions non bornées pour certaines equations quasi-linéaires, Port. Math., 41, 507-534 (1982; Zbl 0544.35040)], which simply relies on the Fatou lemma combined with the sign assumption on \(g\). Cited in 24 Documents MSC: 35J65 Nonlinear boundary value problems for linear elliptic equations 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35A10 Cauchy-Kovalevskaya theorems 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems Keywords:linear growth; existence; weak solution Citations:Zbl 0544.35040 PDFBibTeX XMLCite \textit{A. Porretta}, Port. Math. 57, No. 2, 179--190 (2000; Zbl 0963.35068) Full Text: EuDML