On the fractional Musielak-Sobolev spaces in Rd: Embedding results & applications

This paper deals with new continuous and compact embedding theorems for the fractional Musielak-Sobolev spaces in Rd. As an application, using the variational methods, we obtain the existence of a nontrivial weak solution for the following Schr & ouml;dinger equation (-Delta)s gx,y u + V(x)g(x, x, u) = b(x)|u|p(x)-2u, for all x is an element of Rd, where (-Delta)sgx,y is the fractional Museilak gx,y-Laplacian, V is a potential function, b is an element of L delta'(x)(Rd), and p, delta is an element of C (Rd, (1, +infinity))boolean AND L infinity(Rd). We would like to mention that the theory of the fractional Musielak-Sobolev spaces is in a developing state and there are few papers on this topic, see [6,11,12]. Note that, all these latter works dealt with the bounded case and there are no results devoted for the fractional MusielakSobolev spaces in Rd. Since the embedding results are crucial in applying variational methods, this work will provide a bridge between the fractional Mueislak-Sobolev theory and PDE's. (c) 2024 Elsevier Inc. All rights reserved.