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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 fractional concave–convex problem
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Fecha
2021
Autores
Ledesma C.E.T.
Bonilla M.C.M.
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Birkhauser
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Abstracto
A new fractional function space EL?[a,b] with Riemann–Liouville fractional derivative and its related properties are established in this paper. Under this configuration, the following fractional concave–convex problem: xDb?(aDx?u)=?u?+up,in(a,b)B?(u)=0,in?(a,b)where ?? (0 , 1) , ?? (0 , 1) and p?(1,1+2?1-2?) if ??(0,12) and p? (1 , + ?) if ??(12,1). B?(u) represent the boundary condition of the problem which depend of the behavior of ?? (0 , 1) , that is: B?(u)={limx?a+aIx1-?u(x)=0,if??(0,12)u(a)=u(b)=0,if??(12,1).By using Ekeland’s variational principle and mountain pass theorem we show that the problem (0.1) at less has two nontrivial weak solutions. © 2021, Tusi Mathematical Research Group (TMRG).
Descripción
This work was partially supported by CONCYTEC PERU, 379-2019-FONDECYT “ASPECTOS CUALITATIVOS DE ECUACIONES NO-LOCALES Y APLICACIONES”
Palabras clave
Variational methods,
Fractional Riemman–Liouville operators,
Fractional Sobolev spaces,
Nonlocal problems