Abstract

In this paper, we give some properties of the new fractional Sobolev spaces with variable exponents and apply them to study a class of eigenvalue problems involving the fractional p(·)-Laplace operator. We obtain sequences of eigenvalues going asymptotically to infinity and we also establish sufficient conditions to get zero value for the principal eigenvalue, which is a striking difference between the variable exponent case and the constant exponent case. As an application, we obtain several existence and nonexistence results for the eigenvalue problem according to the asymptotic growth of the nonlinearity and the range of the spectral parameter.