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Existence and uniqueness results for solutions of nonlinear equations with right hand side in \(L^1\). (English) Zbl 0891.35039

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.

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
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