Fractional Sobolev space with Riemann-Liouville fractional derivative and application to a fractional concave-convex problem. (English) Zbl 1490.26006
This paper deals with the existence of weak solutions to a class of Riemann-Liouville fractional equations. In order to prove the existence, the authors introduce a Sobolev space, in which the solutions lie, and provide several properties of this space so that classic arguments for the existence of weak solutions can be employed. In fact, 2/3 of the paper is devoted to obtaining such properties, most of which seem to have a limited novelty in the reviewer’s opinion.
The main contribution of this paper is proving the existence of weak solutions for equations with small derivation order. In the authors’, opinion this is due to the weak character of the proposed solutions and the introduced Sobolev space, suggesting that their approach can be followed to prove the solvability of other fractional equations with small derivation order. Likely to be true, the reader must consider, however, that the introduced space is a fractional Sobolev space and the special form of the equation.
The paper is in general well-written (there are some annoying typos) and technically sound.
The main contribution of this paper is proving the existence of weak solutions for equations with small derivation order. In the authors’, opinion this is due to the weak character of the proposed solutions and the introduced Sobolev space, suggesting that their approach can be followed to prove the solvability of other fractional equations with small derivation order. Likely to be true, the reader must consider, however, that the introduced space is a fractional Sobolev space and the special form of the equation.
The paper is in general well-written (there are some annoying typos) and technically sound.
Reviewer: Javier Gallegos (Santiago de Chile)
MSC:
| 26A33 | Fractional derivatives and integrals |
| 34B15 | Nonlinear boundary value problems for ordinary differential equations |
| 35A15 | Variational methods applied to PDEs |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
Keywords:
variational methods; fractional Riemann-Liouville operators; nonlocal problems; fractional Sobolev spacesReferences:
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