Abstract
The class of problems treated here are elliptic partial differential equations with a homogeneous boundary condition and a non-linear perturbation obtained by composition with a fixed smooth function. The existence of solutions is obtained from the Leray--Schauder theorem or under a Landesman--Lazer condition on the data. Existence is carried over to a wide range of $L_p$-Sobolev spaces, using a non-trivial procedure to obtain a general regularity result. In fact the results are obtained in the general scales of Besov and Triebel--Lizorkin spaces.
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