Fiorenza, A.; Sbordone, C. Existence and uniqueness results for solutions of nonlinear equations with right hand side in \(L^1\). (English) Zbl 0891.35039 Stud. Math. 127, No. 3, 223-231 (1998). Summary: We prove an existence and uniqueness theorem for the elliptic Dirichlet problem for the equation \(\text{div }a(x,\nabla u)= f\) in a planar domain \(\Omega\). Here \(f\in L^1(\Omega)\) and the solution belongs to the so-called grand Sobolev space \(W^{(1,2)}_0(\Omega)\). This is the proper space when the right-hand side is assumed to be only \(L^1\)-integrable. In particular, we obtain the exponential integrability of the solution, which in the linear case was previously proved by Brezis-Merle and Chanillo-Li. Cited in 71 Documents MSC: 35J65 Nonlinear boundary value problems for linear elliptic equations 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 35R05 PDEs with low regular coefficients and/or low regular data Keywords:grand Sobolev space; exponential integrability PDFBibTeX XMLCite \textit{A. Fiorenza} and \textit{C. Sbordone}, Stud. Math. 127, No. 3, 223--231 (1998; Zbl 0891.35039) Full Text: EuDML