A class of elliptic inclusion in fractional Orlicz–Sobolev spaces

ABSTRACT

We consider an elliptic inclusion that is driven by $(-\mathrm{\Delta }{)}_{\mathrm{a}(\cdot )}^s$ called fractional $\mathrm{a}(\cdot )$-Laplacian operator, with Dirichlet-type boundary conditions. We take two different assumptions on the Carathéodory function $\mathrm{f}$. One of them is where $\mathrm{f}$ satisfies the localy Lipchitz condition and the other one is when $\mathrm{f}$ does not satisfy an assumption of growth. With a regular Clarke subdifferential ${\mathrm{\partial }}_C$ and Lebourg's mean value theorem, by applying a variational approach together with the critical point theory, we obtain a weak solution to the inclusion problem in appropriate fractional Orlicz–Sobolev spaces.

KEYWORDS:

• Fractional Orlicz–Sobolev spaces
• elliptic inclusion
• variational approach
• critical point theory

AMS SUBJECT CLASSIFICATIONS:

• 35J60
• 35J20
• 35S15

Acknowledgments

The author would like to thank the anonymous referee for carefully reading the manuscript and for his/her useful comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).