In this paper we study the asymptotic behavior of the least energy nodal solution of a problem with a jumping nonlinearity.
@article{ASNSP_2011_5_10_1_19_0, author = {Dancer, Edward N. and Santra, Sanjiban and Wei, Juncheng}, title = {Least energy nodal solution of a singular perturbed problem with jumping nonlinearity}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 10}, number = {1}, year = {2011}, pages = {19-36}, zbl = {1219.35103}, mrnumber = {2829319}, language = {en}, url = {http://www.numdam.org/item/ASNSP_2011_5_10_1_19_0} }
Dancer, Edward N.; Santra, Sanjiban; Wei, Juncheng. Least energy nodal solution of a singular perturbed problem with jumping nonlinearity. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 10 (2011) no. 1, pp. 19-36. http://www.numdam.org/item/ASNSP_2011_5_10_1_19_0/
[1] T. Bartsch and T. Weth, A note on additional properties of sign changing solutions to superlinear elliptic equations, Topol. Methods Nonlinear Anal. 22 (2003), 1–14. | MR 2037264 | Zbl 1094.35041
[2] T. Bartsch, T. Weth and M. Willem, Partial symmetry of least energy nodal solutions to some variational problems, J. Anal. Math. 96 (2005), 1–18. | MR 2177179 | Zbl 1206.35086
[3] V. Benci and G. Cerami, Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology, Calc. Var. Partial Differential Equations 2 (1994), 29–48. | MR 1384393 | Zbl 0822.35046
[4] S. Chang, C. S. Lin, T. C. Lin and W. Lin, Segregated nodal domains of two-dimensional multispecies Bose-Einstein condensates, Phys. D 196 (2004), 341–361. | MR 2090357 | Zbl 1098.82602
[5] M. Conti, S. Terracini and G. Verzini, Nehari’s problem and competing species system, Ann. Inst. H. Poincaré Anal. Non Linéaire 19 (2002), 871–888. | Numdam | MR 1939088 | Zbl 1090.35076
[6] M. Conti, S. Terracini and G. Verzini, An optimal partition problem related to nonlinear eigenvalues, J. Funct. Anal. 198 (2003), 160–196. | MR 1962357 | Zbl 1091.35051
[7] M. Conti, S. Terraciniand G. Verzini, Asymptotic estimates for the spatial segregation of competitive systems, Adv. Math. 195 (2005), 524–560. | MR 2146353 | Zbl 1126.35016
[8] B. Noris, H. Tavares, S. Terracini and G. Verzini, Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition, Comm. Pure Appl. Math. 63 (2010), 267–302. | MR 2599456 | Zbl 1189.35314
[9] E. N. Dancer and Y. Du, Competing species equations with diffusion, large interactions, and jumping nonlinearities, J. Differential Equations 114 (1994), 434–475. | MR 1303035 | Zbl 0815.35024
[10] E. N. Dancer, J. C. Wei and 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 | MR 2629888 | Zbl 1191.35121
[11] M. del Pino and P. Felmer, Spike-layered solutions of singularly perturbed elliptic problems in a degenerate setting, Indiana Univ. Math. J. 48 (1999), 883–898. | MR 1736974 | Zbl 0932.35080
[12] M. Esteban and P. Lions, Existence and nonexistence results for semilinear elliptic problems in unbounded domains, Proc. Roy. Soc. Edinburgh Sect. A 93 (1982/83), , 1–14. | MR 688279 | Zbl 0506.35035
[13] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), 209–243. | MR 544879 | Zbl 0425.35020
[14] T.-C. Lin and J.-C. Wei, Spikes in two coupled nonlinear Schrodinger equations, Ann. Inst. H. Poincaré Anal. Nonlinéare 22 (2005), 403–439. | Numdam | MR 2145720 | Zbl 1080.35143
[15] W. M. Ni and J. Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems, Comm. Pure Appl. Math. 48 (1995), 731–768. | MR 1342381 | Zbl 0838.35009
[16] W. M. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math. 44 (1991), 819–851. | MR 1115095 | Zbl 0754.35042
[17] W. M. Ni and I. Takagi Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Math. J. 70 (1993), 247–281. | MR 1219814 | Zbl 0796.35056
[18] Ezzat S. Noussair and J. Wei On the effect of domain geometry on the existence of nodal solutions in singular perturbations problems, Indiana Univ. Math. J. 46 (1997), 1255–1271. | MR 1631584 | Zbl 0907.35011
[19] J. C. Wei and T. Weth, Asymptotic behavior of solutions of planar elliptic systems with strong competition, Nonlinearity 21 (2008), 305–317. | MR 2384550 | Zbl 1132.35482
[20] J. C. Wei and T. Weth, Radial solutions and phase separation in a system of two coupled Schrödinger equations, Arch. Ration. Mech. Anal. 190 (2008), 83–106. | MR 2434901 | Zbl 1161.35051
[21] M. Willem, “Minimax theorems”, Progress in Nonlinear Differential Equations and their Applications, Vol. 24, Birkhäuser Boston, Inc., Boston, MA, 1996. | MR 1400007 | Zbl 0856.49001