In this paper we study the existence of nontrivial solutions for the problem < < py 2 N ⌬ u s u u in a bounded smooth domain ⍀ ; ޒ , with a nonlinear boundary p < < py 2 Ž . condition given by ٌu Ѩ urѨ s f u on the boundary of the domain. The proofs are based on variational and topological arguments.
In this paper we extend the well-known concentration-compactness principle for the Fractional Laplacian operator in unbounded domains. As an application we show sufficient conditions for the existence of solutions to some critical equations involving the fractional p−laplacian in the whole R n .
In this paper we define the notion of nonlocal magnetic Sobolev spaces with nonstandard growth for Lipschitz magnetic fields. In this context we prove a Bourgain -Brezis -Mironescu type formula for functions in this space as well as for sequences of functions. Finally, we deduce some consequences such as the Γ−convergence of modulars and convergence of solutions for some non-local magnetic Laplacian allowing non-standard growth laws to its local counterpart.On the other hand, when studying phenomena allowing behaviors more general than power laws, such as anisotropic fluids with flows obeying nonstandard rheology [8,17] or capillarity phenomena, 2010 Mathematics Subject Classification. 46E30, 35R11, 45G05.
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