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Sobolev embeddings with variable exponent

Eigenvalues of p(x)-Laplacian Dirichlet problem

Global C1,α regularity for variable exponent elliptic equations in divergence form

We consider the Hardy–Littlewood maximal operator M on Musielak–Orlicz Spaces L φ ( R d ) . We give a necessary condition for the continuity of M on L φ ( R d ) which generalizes the concept of Muckenhoupt classes. In the special case of generalized Lebesgue spaces L p ( ⋅ ) ( R d ) we show that this condition is also sufficient. Moreover, we show that the condition is “left-open” in the sense that not only M but also M q is continuous for some q > 1 , where M q f = ( M ( | f | q ) ) 1 q .

Maximal function on Musielak–Orlicz spaces and generalized Lebesgue spaces

We study the boundary value problemKdivðaðx;VuÞÞZlðu gK1 Ku bK1 Þ in U, uZ 0o nvU, where U is a smooth bounded domain in R N and divðaðx;VuÞÞ is a pðxÞ-Laplace type operator, with 1!b!g! infx2UpðxÞ. We prove that if l is large enough then there exist at least two non-negative weak solutions. Our approach relies on the variable exponent theory of generalized Lebesgue–Sobolev spaces, combined with adequate variational methods and a variant of the Mountain Pass lemma.

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A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids

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Jiří Rákosník

Papers

On spaces $L^{p(x)}$ and $W^{k, p(x)}$