Ground state solution for the Choquard equation under the superposition of operators of mixed fractional order

Abstract

We establish the existence of a ground state solution for the fractional Choquard equation governed by the superposition operator ∫[0,1](-Δ)sudμ(s),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \int _{[0, 1]} (-\varDelta )^s u\, d\mu (s), $$\end{document}and in presence of a confining potential. Here, μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} denotes a signed measure on the interval of fractional exponents [0, 1]. The presence of the superposition operator asks for the problem to be addressed with a special care. However, the wide generality of this setting allows to provide entirely new existence results in several special cases of interest, including, e.g., the mixed operator one -Δ+(-Δ)s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\varDelta + (-\varDelta )^s$$\end{document}. We point out that the possibility of considering operators “with the wrong sign" is also a complete novelty.