On spaces \(L^{p(x)}\) and \(W^{k,p(x)}\). (English) Zbl 0784.46029

The spaces mentioned in the title are certain versions of the Lebesgue and the Sobolev spaces with variable order of integrability. The authors establish elementary properties of these spaces and prove some embedding theorems for Sobolev-type spaces. Then they apply the results to the proof of existence of a weak solution of a certain nonlinear boundary value problem.


46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35J65 Nonlinear boundary value problems for linear elliptic equations
35D30 Weak solutions to PDEs
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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