Sign-changing solutions of competition–diffusion elliptic systems and optimal partition problems

Tavares, Hugo ; Terracini, Susanna

Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012), p. 279-300 / Harvested from Numdam

Résumé

Dans cet article nous montrons lʼexistence dʼune infinité de solutions qui changent de signe pour le système dʼéquations de Schrödinger avec des interactions compétitives -Δu i +a i u i 3 +βu i ∑ j≠i u j 2 =λ i,β u i ,u i ∈H 0 1 (Ω),i=1,⋯,m où Ω est un domaine borné, $ \beta >0$ et $ {a}_{i}⩾0$ ∀i. De plus, quand $ {a}_{i}=0$, nous démontrons une relation entre les énergies critiques associées à ce système et le problème de partition optimale inf ω i ⊂Ωopen ω i ∩ω j =⌀∀i≠j∑ i=1 m λ k i (ω i ), où $ {\lambda }_{{k}_{i}}(\omega )$ indiques la $ {k}_{i}$-ème valeur propre de lʼopérateur −Δ in $ {H}_{0}^{1}(\omega )$. Dans le cas $ {k}_{i}⩽2$, nous montrons que le problème de partition optimale apparaît comme une valeur limite critique, en tant que paramètre de compétition β diverge vers +∞.

Abstract

In this paper we prove the existence of infinitely many sign-changing solutions for the system of m Schrödinger equations with competition interactions -Δu i +a i u i 3 +βu i ∑ j≠i u j 2 =λ i,β u i ,u i ∈H 0 1 (Ω),i=1,⋯,m where Ω is a bounded domain, $ \beta >0$ and $ {a}_{i}⩾0$ ∀i. Moreover, for $ {a}_{i}=0$, we show a relation between critical energies associated with this system and the optimal partition problem inf ω i ⊂Ωopen ω i ∩ω j =⌀∀i≠j∑ i=1 m λ k i (ω i ), where $ {\lambda }_{{k}_{i}}(\omega )$ denotes the $ {k}_{i}$-th eigenvalue of −Δ in $ {H}_{0}^{1}(\omega )$. In the case $ {k}_{i}⩽2$ we show that the optimal partition problem appears as a limiting critical value, as the competition parameter β diverges to +∞.

Détail

Publié le : 2012-01-01

MR 2901198 | Zbl 1241.35046 | 1 citation dans Numdam

DOI : https://doi.org/10.1016/j.anihpc.2011.10.006

Bibliographie

[1] T. Bartsch, N. Dancer, Z.-Q. Wang, A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system, Calc. Var. Partial Differential Equations 37 (2010), 345-361 | MR 2592975 | Zbl 1189.35074

[2] T. Bartsch, Z. Liu, T. Weth, Sign changing solutions of superlinear Schrödinger equations, Comm. Partial Differential Equations 29 (2004), 25-42 | MR 2038142 | Zbl 1140.35410

[3] B. Bourdin, D. Bucur, E. Oudet, Optimal partitions for eingenvalues, J. Sci. Comp. 6 (2009), 4100-4114 | MR 2566585 | Zbl 1207.49050

[4] D. Bucur, G. Buttazzo, A. Henrot, Existence results for some optimal partition problems, Adv. Math. Sci. Appl. 8 (1998), 571-579 | MR 1657219 | Zbl 0915.49006

[5] L.A. Caffarelli, F.-H. Lin, Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries, J. Amer. Math. Soc. 21 (2008), 847-862 | MR 2393430 | Zbl 1194.35138

[6] L.A. Caffarelli, F.-H. Lin, An optimal partition problem for eigenvalues, J. Sci. Comp. 31 (2007), 5-18 | MR 2304268 | Zbl 1123.65060

[7] S. Chang, C. Lin, T. Lin, W. Lin, Segregated nodal domains of two-dimensional multispecies Bose–Einstein condensates, Phys. D 196 (2004), 341-361 | MR 2090357 | Zbl 1098.82602

[8] K.-C. Chang, Z.-Q. Wang, T. Zhang, On a new index theory and non semi-trivial solutions for elliptic systems, Discrete Contin. Dyn. Syst. 28 (2010), 809-826 | MR 2644766 | Zbl 1193.35036

[9] M. Conti, S. Terracini, G. Verzini, On a class of optimal partition problems related to the Fucik spectrum and to the monotonicity formula, Calc. Var. Partial Differential Equations 22 (2005), 45-72 | MR 2105968 | Zbl 1132.35365

[10] M. Conti, L. Merizzi, S. Terracini, Remarks on variational methods and lower-upper solutions, NoDEA Nonlinear Differential Equations Appl. 6 (1999), 371-393 | MR 1736543 | Zbl 0941.35031

[11] N. Dancer, J.-C. Wei, T. Weth, A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system, Ann. Inst. H. Poincaré Anal. Non Linéaire 27 (2010), 953-969 | Numdam | Zbl 1191.35121

[12] A. Domingos, M. Ramos, Existence and bounds for nonlinear Schrödinger systems with strong competition, preprint, 2011.

[13] Z. Dzedzej, A. Idzik, M. Izydorek, Borsuk–Ulam type theorems on product spaces I, Bull. Polish Acad. Sci. Math. 48 (2000), 379-386 | Zbl 0968.55001

[14] Z. Dzedzej, A. Idzik, M. Izydorek, Borsuk–Ulam type theorems on product spaces II, Topol. Methods Nonlinear Anal. 14 (1999), 345-352 | Zbl 0961.47032

[15] B. Helffer, T. Hoffmann-Ostenhof, S. Terracini, Nodal domains and spectral minimal partitions, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), 101-138 | Numdam | Zbl 1171.35083

[16] M. Ramos, H. Tavares, W. Zou, A Bahri–Lions theorem revisited, Adv. Math. 222 (2009), 2173-2195 | Zbl 1182.35118

[17] B. Noris, M. Ramos, Existence and bounds of positive solutions for a nonlinear Schrödinger system, Proc. Amer. Math. Soc. 138 (2010), 1681-1692 | Zbl 1189.35086

[18] B. Noris, H. Tavares, S. Terracini, G. Verzini, Convergence of minimax and continuation of critical points for singularly perturbed systems, J. Eur. Math. Soc., in press.

[19] B. Noris, H. Tavares, S. Terracini, G. Verzini, Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition, Comm. Pure Appl. Math. 63 (2010), 267-302 | Zbl 1189.35314

[20] H. Tavares, S. Terracini, Regularity of the nodal set of segregated critical configurations, arXiv:1002.3822 (2010)

[21] J.-C. Wei, T. Weth, Asymptotic behaviour of solutions of planar elliptic systems with strong competition, Nonlinearity 21 (2008), 305-317 | Zbl 1132.35482

[22] C. Zhong, The Borsuk–Ulam theorem on product space, K.K. Tan (ed.), Fixed Point Theory and Applications, World Sci. Publ., River Edge, NJ (1992), 362-372