A critical fractional equation with concave–convex power nonlinearities
Annales de l'I.H.P. Analyse non linéaire, Volume 32 (2015) no. 4, pp. 875-900.

In this work we study the following fractional critical problem

(P λ )={(-Δ) s u=λu q +u 2 s -1 ,u>0inΩ,u=0in n Ω,
where Ω n is a regular bounded domain, λ>0, 0<s<1 and n>2s. Here (-Δ) s denotes the fractional Laplace operator defined, up to a normalization factor, by
-(-Δ) s u(x)= n u(x+y)+u(x-y)-2u(x) |y| n+2s dy,x 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.

DOI: 10.1016/j.anihpc.2014.04.003
Classification: 49J35,  35A15,  35S15,  47G20,  45G05
Keywords: Fractional Laplacian, Critical nonlinearities, Convex–concave nonlinearities, Variational techniques, Mountain Pass Theorem
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     title = {A critical fractional equation with concave{\textendash}convex power nonlinearities},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {875--900},
     publisher = {Elsevier},
     volume = {32},
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}
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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, Volume 32 (2015) no. 4, pp. 875-900. doi : 10.1016/j.anihpc.2014.04.003. http://archive.numdam.org/articles/10.1016/j.anihpc.2014.04.003/

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