Geodesic
On the asymptotic growth of positive solutions to a nonlocal elliptic blow-up system involving strong competition
Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 3, pp. 831-858 Cet article a éte moissonné depuis la source Numdam

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For a competition-diffusion system involving the fractional Laplacian of the form

−(−Δ)su=uv2,−(−Δ)sv=vu2,u,v>0inRN,
with s∈(0,1), we prove that the maximal asymptotic growth rate for its entire solutions is 2s. Moreover, since we are able to construct symmetric solutions to the problem, when N=2 with prescribed growth arbitrarily close to the critical one, we can conclude that the asymptotic bound found is optimal. Finally, we prove existence of genuinely higher dimensional solutions, when N≥3. Such problems arise, for example, as blow-ups of fractional reaction-diffusion systems when the interspecific competition rate tends to infinity.

DOI : 10.1016/j.anihpc.2017.08.004
Classification : 35J65, 35B40, 35B44, 35R11, 81Q05, 82B10
Keywords: Fractional Laplacian, Spatial segregation, Strongly competing systems, Entire solutions 
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     author = {Terracini, Susanna and Vita, Stefano},
     title = {On the asymptotic growth of positive solutions to a nonlocal elliptic blow-up system involving strong competition},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {831--858},
     year = {2018},
     publisher = {Elsevier},
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     number = {3},
     doi = {10.1016/j.anihpc.2017.08.004},
     mrnumber = {3778654},
     zbl = {1393.35042},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2017.08.004/}
}
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Terracini, Susanna; Vita, Stefano. On the asymptotic growth of positive solutions to a nonlocal elliptic blow-up system involving strong competition. Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 3, pp. 831-858. doi: 10.1016/j.anihpc.2017.08.004

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