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Existence and uniqueness results for solutions of nonlinear equations with right hand side in L 1

A. FiorenzaC. Sbordone — 1998

Studia Mathematica

We prove an existence and uniqueness theorem for the elliptic Dirichlet problem for the equation div a(x,∇u) = f in a planar domain Ω. Here f L 1 ( Ω ) and the solution belongs to the so-called grand Sobolev space W 0 1 , 2 ) ( Ω ) . 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.

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