# Fractional Sobolev space with Riemann–Liouville fractional derivative and application to a fractional concave–convex problem

Fractional Sobolev space with Riemann–Liouville fractional derivative and application to a... A new fractional function space ELα[a,b]\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$E_{L}^{\alpha }[a,b]$$\end{document} with Riemann–Liouville fractional derivative and its related properties are established in this paper. Under this configuration, the following fractional concave–convex problem: 0.1xDbα(aDxαu)=λuσ+up,in(a,b)Bα(u)=0,in∂(a,b)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}\begin{aligned} \begin{aligned}&{_{x}}D_{b}^{\alpha }({_{a}}D_{x}^{\alpha }u) = \lambda u^\sigma + u^p,\;\;\text{ in }\;\;(a,b)\\&B_{\alpha }(u)=0,\;\;\text{ in }\;\;\partial (a,b) \end{aligned} \end{aligned}\end{document}where α∈(0,1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha \in (0,1)$$\end{document}, σ∈(0,1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\sigma \in (0,1)$$\end{document} and p∈(1,1+2α1-2α)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p\in (1, \frac{1+ 2\alpha }{1-2\alpha })$$\end{document} if α∈(0,12)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha \in (0, \frac{1}{2})$$\end{document} and p∈(1,+∞)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p\in (1, +\infty )$$\end{document} if α∈(12,1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha \in (\frac{1}{2}, 1)$$\end{document}. Bα(u)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$B_\alpha (u)$$\end{document} represent the boundary condition of the problem which depend of the behavior of α∈(0,1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha \in (0,1)$$\end{document}, that is: Bα(u)=limx→a+aIx1-αu(x)=0,ifα∈0,12u(a)=u(b)=0,ifα∈12,1.\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}\begin{aligned} B_\alpha (u) = {\left\{ \begin{array}{ll} \lim _{x\rightarrow a^+}{_{a}}I_{x}^{1-\alpha }u(x) = 0,&{}\hbox { if}\ \alpha \in \left(0, \frac{1}{2}\right)\\ u(a) = u(b) = 0,&{}\hbox { if}\ \alpha \in \left(\frac{1}{2}, 1\right). \end{array}\right. } \end{aligned}\end{document}By using Ekeland’s variational principle and mountain pass theorem we show that the problem (0.1) at less has two nontrivial weak solutions. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Operator Theory Springer Journals

# Fractional Sobolev space with Riemann–Liouville fractional derivative and application to a fractional concave–convex problem

, Volume 6 (4) – Oct 1, 2021
38 pages

/lp/springer-journals/fractional-sobolev-space-with-riemann-liouville-fractional-derivative-qscWABpmy7
Publisher
Springer Journals
ISSN
2662-2009
eISSN
2538-225X
DOI
10.1007/s43036-021-00159-w
Publisher site
See Article on Publisher Site

### Abstract

A new fractional function space ELα[a,b]\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$E_{L}^{\alpha }[a,b]$$\end{document} with Riemann–Liouville fractional derivative and its related properties are established in this paper. Under this configuration, the following fractional concave–convex problem: 0.1xDbα(aDxαu)=λuσ+up,in(a,b)Bα(u)=0,in∂(a,b)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}\begin{aligned} \begin{aligned}&{_{x}}D_{b}^{\alpha }({_{a}}D_{x}^{\alpha }u) = \lambda u^\sigma + u^p,\;\;\text{ in }\;\;(a,b)\\&B_{\alpha }(u)=0,\;\;\text{ in }\;\;\partial (a,b) \end{aligned} \end{aligned}\end{document}where α∈(0,1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha \in (0,1)$$\end{document}, σ∈(0,1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\sigma \in (0,1)$$\end{document} and p∈(1,1+2α1-2α)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p\in (1, \frac{1+ 2\alpha }{1-2\alpha })$$\end{document} if α∈(0,12)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha \in (0, \frac{1}{2})$$\end{document} and p∈(1,+∞)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p\in (1, +\infty )$$\end{document} if α∈(12,1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha \in (\frac{1}{2}, 1)$$\end{document}. Bα(u)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$B_\alpha (u)$$\end{document} represent the boundary condition of the problem which depend of the behavior of α∈(0,1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha \in (0,1)$$\end{document}, that is: Bα(u)=limx→a+aIx1-αu(x)=0,ifα∈0,12u(a)=u(b)=0,ifα∈12,1.\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}\begin{aligned} B_\alpha (u) = {\left\{ \begin{array}{ll} \lim _{x\rightarrow a^+}{_{a}}I_{x}^{1-\alpha }u(x) = 0,&{}\hbox { if}\ \alpha \in \left(0, \frac{1}{2}\right)\\ u(a) = u(b) = 0,&{}\hbox { if}\ \alpha \in \left(\frac{1}{2}, 1\right). \end{array}\right. } \end{aligned}\end{document}By using Ekeland’s variational principle and mountain pass theorem we show that the problem (0.1) at less has two nontrivial weak solutions.