Problems involving the fractional \(g\)-Laplacian with lack of compactness. (English) Zbl 1511.46022
Summary: In this paper, we prove compact embedding of a subspace of the fractional Orlicz-Sobolev space \(W^{s, G}\left(\mathbb{R}^N\right)\) consisting of radial functions; our target embedding spaces are of Orlicz type. In addition, we prove a Lions and Lieb type results for \(W^{s, G}\left(\mathbb{R}^N\right)\) that works together in a particular way to get a sequence whose weak limit is non-trivial. As an application, we study the existence of solutions to quasilinear elliptic problems in the whole space \(\mathbb{R}^N\) involving the fractional \(g\)-Laplacian operator, where the conjugated function \(\widetilde{G}\) of \(G\) does not satisfy the \(\Delta_2\)-condition.
©2023 American Institute of Physics
©2023 American Institute of Physics
MSC:
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
| 35R11 | Fractional partial differential equations |
| 26A33 | Fractional derivatives and integrals |
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