On the fractional elliptic problems with difference in the Orlicz-Sobolev spaces

Abstract

In this paper, we deal with a family of the fractional elliptic operators with difference in the Orlicz-Sobolev spaces under boundary and initial conditions. We get a theorem which shows existence of a sequence of weak solutions for the fractional elliptic problems with difference in the Orlicz-Sobolev spaces. We first show that there exists a sequence of weak solutions for this problem on the finite-dimensional subspace. We next show that there exists a limit sequence of the sequence of weak solutions for finite-dimensional problem and this limit sequence is the sequence of the solutions of our problem. We get this result by the estimate of the energy functional and the compactness property of the continuous embedding inclusions between some special spaces.