We obtain fundamental imbeddings for the fractional Sobolev space with variable exponent that is a generalization of well-known fractional Sobolev spaces. As an application, we obtain a-priori bounds and multiplicity of solutions to some nonlinear elliptic problems involving the fractional $p(\cdot)$-Laplacian.
On multiplicity of eigenvalues and symmetry of eigenfunctions of the $p$-Laplacian
Nonlinear fractional magnetic Schrödinger equation: existence and multiplicity
Symmetry and multiplicity of solutions in a two-dimensional Landau-de Gennes model for liquid crystals
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