no. 3
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.

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
Mots clés : Fractional Laplacian, Spatial segregation, Strongly competing systems, Entire solutions 
@article{AIHPC_2018__35_3_831_0,
     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},
     publisher = {Elsevier},
     volume = {35},
     number = {3},
     year = {2018},
     doi = {10.1016/j.anihpc.2017.08.004},
     mrnumber = {3778654},
     zbl = {1393.35042},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2017.08.004/}
}
TY  - JOUR
AU  - Terracini, Susanna
AU  - Vita, Stefano
TI  - On the asymptotic growth of positive solutions to a nonlocal elliptic blow-up system involving strong competition
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2018
SP  - 831
EP  - 858
VL  - 35
IS  - 3
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.anihpc.2017.08.004/
DO  - 10.1016/j.anihpc.2017.08.004
LA  - en
ID  - AIHPC_2018__35_3_831_0
ER  - 
%0 Journal Article
%A Terracini, Susanna
%A Vita, Stefano
%T On the asymptotic growth of positive solutions to a nonlocal elliptic blow-up system involving strong competition
%J Annales de l'I.H.P. Analyse non linéaire
%D 2018
%P 831-858
%V 35
%N 3
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/j.anihpc.2017.08.004/
%R 10.1016/j.anihpc.2017.08.004
%G en
%F AIHPC_2018__35_3_831_0
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. http://archive.numdam.org/articles/10.1016/j.anihpc.2017.08.004/

[1] Berestycki, H.; Lin, T.C.; Wei, J.; Zhao, C. On phase-separation model: asymptotics and qualitative properties, Arch. Ration. Mech. Anal., Volume 208 (2013), pp. 163–200 | DOI | MR | Zbl

[2] Berestycki, H.; Terracini, S.; Wang, K.; Wei, J. Existence and stability of entire solutions of an elliptic system modeling phase separation, Adv. Math., Volume 243 (2013), pp. 102–126

[3] Brock, F.; Solynin, A. An approach to symmetrization via polarization, Trans. Am. Math. Soc., Volume 352 (2000), pp. 1759–1796 | MR | Zbl

[4] Cabré, X.; Sire, Y. Nonlinear equations for fractional Laplacians, I: regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 31 (2014) no. 1 | DOI | Numdam | MR | Zbl

[5] Caffarelli, L.; Silvestre, L. An extension problem related to the fractional Laplacian, Commun. Partial Differ. Equ., Volume 32 (2007) no. 8, pp. 1245–1260 | DOI | MR | Zbl

[6] Colorado, E.; Peral, I. Semilinear elliptic problems with mixed Dirichlet–Neumann boundary conditions, J. Funct. Anal., Volume 199 (2003) no. 2, pp. 468–507 | DOI | MR | Zbl

[7] Farina, A. Some symmetry results for entire solutions of an elliptic system arising in phase separation, Discrete Contin. Dyn. Syst., Volume 34 (2014) no. 6, pp. 2505–2511 | DOI | MR | Zbl

[8] Farina, A.; Soave, N. Monotonicity and 1-dimensional symmetry for solutions of an elliptic system arising in Bose–Einstein condensation, Arch. Ration. Mech. Anal., Volume 213 (2014) no. 1, pp. 287–326 | DOI | MR | Zbl

[9] Noris, B.; Terracini, S.; Tavares, H.; Verzini, G. Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition, Commun. Pure Appl. Math., Volume 63 (2010), pp. 267–302 | DOI | MR | Zbl

[10] Opic, B.; Kufner, A. Hardy-Type Inequalities, Pitman Research Notes in Mathematics Series, vol. 219, Longman Scientific & Technical, Harlow, 1990 | MR | Zbl

[11] Rüland, A. Unique continuation for fractional Schrödinger equations with rough potential, 2013 (preprint) | arXiv | MR

[12] Smets, D.; Willem, M. Partial symmetry and asymptotic behavior for some elliptic variational problems, Calc. Var. Partial Differ. Equ., Volume 18 (2003), pp. 57–75 | DOI | MR | Zbl

[13] Soave, N.; Terracini, S. Liouville theorems and 1-dimensional symmetry for solutions of an elliptic system modelling phase separation, Adv. Math., Volume 279 (2015), pp. 29–66 | DOI | MR | Zbl

[14] Soave, N.; Zilio, A. Entire solutions with exponential growth for an elliptic system modeling phase-separation, Nonlinearity, Volume 27 (2014) no. 2, pp. 305–342 | DOI | MR | Zbl

[15] Soave, N.; Zilio, A. Uniform bounds for strongly competing systems: the optimal Lipschitz case, Arch. Ration. Mech. Anal., Volume 218 (2015) no. 2, pp. 647–697 | DOI | MR | Zbl

[16] Soave, N.; Zilio, A. Multidimensional entire solutions for an elliptic system modelling phase separation, Anal. PDE, Volume 9 (2016) no. 5, pp. 1019–1041 | DOI | MR | Zbl

[17] Soave, N.; Zilio, A. On phase separation in systems of coupled elliptic equations: asymptotic analysis and geometric aspects, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 34 (2017) no. 3, pp. 625–654 | DOI | Numdam | MR | Zbl

[18] Terracini, S.; Verzini, G.; Zilio, A. Uniform Hölder bounds for strongly competing systems involving the square root of the Laplacian, J. Eur. Math. Soc., Volume 18 (2016) no. 12, pp. 2865–2924 | DOI | MR | Zbl

[19] Terracini, S.; Verzini, G.; Zilio, A. Uniform Hölder regularity with small exponent in competition-fractional diffusion systems, Discrete Contin. Dyn. Syst., Volume 34 (2014) no. 6, pp. 2669–2691 | DOI | MR | Zbl

[20] Wang, K. On the De Giorgi type conjecture for an elliptic system modeling phase separation, Commun. Partial Differ. Equ., Volume 39 (2014) no. 4, pp. 696–739 | MR | Zbl

[21] Wang, K.; Wei, J. On the uniqueness of solutions of an nonlocal elliptic system, Math. Ann., Volume 365 (2016) no. 1, pp. 105–153 | MR

Cité par Sources :