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Sobolev spaces with variable exponents on complete manifolds. (English) Zbl 1346.46026

J. Funct. Anal. 270, No. 4, 1379-1415 (2016); corrigendum ibid. 272, No. 3, 1296-1299 (2017).
This paper may be considered as a supplement to the previous paper by the first two authors [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 92, 47–59 (2013; Zbl 1329.46033)], where variable exponent Sobolev spaces on Riemann manifolds have already been studied. In the present manuscript, variable exponent spaces of Lebesgue, Sobolev and Hölder type are investigated on complete non-compact Riemann manifolds.
Under suitable assumptions on the geometry of the manifold \(M\) and the \(\log\)-Hölder continuity of the (variable) exponent \(p(\cdot)\), Sobolev embeddings into variable Lebesgue spaces (in the case \(\mathrm{ess }\mathrm{supp}(x)<n\)) and into variable order Hölder spaces (when \(\mathrm{ess }\inf p(x)>n\)) are obtained. Moreover, compact embeddings for \(H\)-invariant Sobolev subspaces (where e.g. \(H\) stands for a compact Lie subgroup of the group of isometries on \(M\)) are also derived. The authors also apply their results to study the existence of weak solutions to non-homogeneous elliptic PDEs involving the \(p(x)\)-Laplacian.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
53B21 Methods of local Riemannian geometry
58J05 Elliptic equations on manifolds, general theory

Citations:

Zbl 1329.46033
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References:

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