A critical fractional equation with concave–convex power nonlinearities

Barrios, B.; Colorado, E.; Servadei, R.; Soria, F.

Barrios, B. ; Colorado, E. ; Servadei, R. ; Soria, F.

Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015), p. 875-900 / Harvested from Numdam

Résumé

In this work we study the following fractional critical problem $(P_{λ}) = {\begin{matrix} {(- Δ)}^{s} u = λ u^{q} + u^{2_{s}^{⁎} - 1}, u > 0 & in Ω, \\ u = 0 & in ℝ^{n} ∖ Ω, \end{matrix}$ where $Ω \subset ℝ^{n}$ is a regular bounded domain, $λ > 0$ , $0 < s < 1$ and $n > 2 s$ . Here ${(- Δ)}^{s}$ denotes the fractional Laplace operator defined, up to a normalization factor, by $- {(- Δ)}^{s} u (x) = \int_{ℝ^{n}} \frac{u (x + y) + u (x - y) - 2 u (x)}{{| y |}^{n + 2 s}} d y, x \in ℝ^{n} .$ Our main results show the existence and multiplicity of solutions to problem $(P_{λ})$ for different values of λ. The dependency on this parameter changes according to whether we consider the concave power case ( $0 < q < 1$ ) or the convex power case ( $1 < q < 2_{s}^{⁎} - 1$ ). These two cases will be treated separately.

Publié le : 2015-01-01

MR 3390088 | Zbl 1350.49009

DOI : https://doi.org/10.1016/j.anihpc.2014.04.003
Classification: 49J35, 35A15, 35S15, 47G20, 45G05

@article{AIHPC_2015__32_4_875_0,
     author = {Barrios, B. and Colorado, E. and Servadei, R. and Soria, F.},
     title = {A critical fractional equation with concave--convex power nonlinearities},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {32},
     year = {2015},
     pages = {875-900},
     doi = {10.1016/j.anihpc.2014.04.003},
     mrnumber = {3390088},
     zbl = {1350.49009},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2015__32_4_875_0}
}

Barrios, B.; Colorado, E.; Servadei, R.; Soria, F. A critical fractional equation with concave–convex power nonlinearities. Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) pp. 875-900. doi : 10.1016/j.anihpc.2014.04.003. http://gdmltest.u-ga.fr/item/AIHPC_2015__32_4_875_0/

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