Fractional Musielak-Sobolev spaces: study of generalized double phase problem with Choquard-logarithmic nonlinearity

Abstract

In this investigation, we conduct a rigorous analysis of a class of non-homogeneous generalized double phase problems, characterized by the inclusion of the fractional \(\phi _{x ,y}^i(\cdot )\)-Laplacian operator (where \(i=1,2\)) and a Choquard-logarithmic nonlinearity, along with a real parameter. Our methodology involves establishing a set of precise conditions related to the Choquard nonlinearities and the continuous function \(\phi _{x ,y}^i\), under which we are able to confirm the existence of multiple distinct solutions to the problem. The analysis is situated within the realm of fractional modular spaces. Key to our approach is the application of the mountain pass theorem, which allows us to circumvent the necessity of the Palais-Smale condition, beside this we lay in the strategic use of the Hardy-Littlewood-Sobolev inequality to underpin the theoretical framework of our study.