Existence and symmetry results for competing variational systems

Nonlinear Differential Equations and Applications

Volumn 20, Issue 3, 2013, Pages 715-740

  Tavares, H ^a   Weth, T ^b  

^a UNIVERSITY OF LISBON   (Portugal)

^b GOETHE UNIVERSITY   (Germany)

Author keywords

Competitive systems; Elliptic gradient systems; Foliated Schwarz symmetry; Ground states solutions; Positive solutions

Indexed keywords

EID: 84878398096     PISSN: 10219722     EISSN: 14209004     Source Type: Journal    
DOI: 10.1007/s00030-012-0176-z     Document Type: Article

References (24)

1. Aftalion A., Pacella F.: Qualitative properties of nodal solutions of semilinear elliptic equations in radially symmetric domains. C. R. Math. Acad. Sci. Paris 339, 339-344 (2004).
2. Ambrosetti A., Rabinowitz P.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349-381 (1973).
3. Bartsch T., Dancer N., Wang Z. Q.: A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system. Calc. Var. Partial Differ. Equ. 37, 345-361 (2010).
4. Brock F.: Symmetry and monotonicity of solutions to some variational problems in cylinders and annuli. Electron. J. Differ. Equ. 108, 20 (2003).
5. Các N. P., Schulz F.: Multiple solutions for a class of semilinear Dirichlet problems. Houston J. Math. 17, 71-81 (1991).
6. Chang S., Lin C., Lin T., Lin W.: Segregated nodal domains of the two-dimensional multispecies Bose-Einstein condensates. Phys. D 196, 341-361 (2004).
7. Conti M., Terracini S., Verzini G.: Nehari's problem and competing species systems. Ann. I. H. Poincaré AN 19, 871-888 (2002).
8. Conti M., Terracini S., Verzini G.: An optimal partition problem related to nonlinear eigenvalues. J. Funct. Anal. 198, 160-196 (2003).
9. Dancer N., Wei J.-C., Weth T.: A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system. Ann. I. H. Poincaré AN 27, 953-969 (2010).
10. Han, Q., Lin, F.: Elliptic partial differential equations. Courant Lecture Notes in Mathematics 1. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence (1997).
11. Lax P. D.: Linear Algebra, Pure and Applied Mathematics. Wiley, New York (1997).
12. Lin T.-C., Wei J.-C.: Spikes in two coupled nonlinear Schrödinger equations. Ann. I. H. Poincaré AN 22, 403-439 (2005).
13. Lin T.-C., Wei J.-C.: Spikes in two-component systems of nonlinear Schrödinger equations with trapping potentials. J. Differ. Equ. 229, 538-569 (2006).
14. Mariş M.: On the symmetry of minimizers. Arch. Ration. Mech. Anal. 192, 311-330 (2009).
15. Noris B., Ramos M.: Existence and bounds of positive solutions for a nonlinear Schrödinger system. Proc. Am. Math. Soc. 138, 1681-1692 (2010).
16. Noris B., Tavares H., Terracini S., Verzini G.: Convergence of minimax and continuation of critical points for singularly perturbed systems. J. Eur. Math. Soc. 14, 1245-1273 (2012).
17. Parkins A. S., Walls D. F.: The physics of trapped dilute-gas Bose-Einstein Condensates. Phys. Rep. 303, 1-80 (1998).
18. Quitner P., Souplet P.: Optimal Liouville-type theorems for noncooperative elliptic Schrödinger systems and applications. Commun. Math. Phys. 311, 1-19 (2012).
19. n. Commun. Math. Phys. 271, 199-221 (2007).
20. Struwe M.: Variational Methods, 2nd edn. Springer, Berlin (1996).
21. Szulkin A., Weth T.: The method of Nehari manifold. In: Gao, D. Y., Motreanu, D. (eds) Handbook of Nonconvex Analysis and Applications. pp. 597-632. International Press, Boston (2010).
22. Tian R., Wang Z. Q.: Multiple wave solutions of nonlinear Schrödinger systems. Top. Methods Nonlinear Anal. 37, 203-223 (2011).
23. Troy W. C.: Symmetry properties in systems of semilinear elliptic equations. J. Differ. Equ. 42, 400-413 (1981).
24. Weth T.: Symmetry of solutions to variational problems for nonlinear elliptic equations via reflection methods. Jahresber. Deutsch. Math.-Ver 112, 119-158 (2010).