Spike-layered solutions for an elliptic system with Neumann boundary conditions

Ramos, Miguel; Yang, Jianfu

doi:10.1090/S0002-9947-04-03659-1

Spike-layered solutions for an elliptic system with Neumann boundary conditions
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by Miguel Ramos and Jianfu Yang PDF

Trans. Amer. Math. Soc. 357 (2005), 3265-3284 Request permission

Abstract:

We prove the existence of nonconstant positive solutions for a system of the form $-\varepsilon ^2\Delta u + u = g(v)$, $-\varepsilon ^2\Delta v + v = f(u)$ in $\Omega$, with Neumann boundary conditions on $\partial \Omega$, where $\Omega$ is a smooth bounded domain and $f$, $g$ are power-type nonlinearities having superlinear and subcritical growth at infinity. For small values of $\varepsilon$, the corresponding solutions $u_{\varepsilon }$ and $v_{\varepsilon }$ admit a unique maximum point which is located at the boundary of $\Omega$.

References

A. Abbondandolo, P. Felmer, and J. Molina, An estimate on the relative Morse index for strongly indefinite functionals, Proceedings of the USA-Chile Workshop on Nonlinear Analysis (Viña del Mar-Valparaiso, 2000) Electron. J. Differ. Equ. Conf., vol. 6, Southwest Texas State Univ., San Marcos, TX, 2001, pp. 1–11. MR 1804760
Alberto Abbondandolo and Juan Molina, Index estimates for strongly indefinite functionals, periodic orbits and homoclinic solutions of first order Hamiltonian systems, Calc. Var. Partial Differential Equations 11 (2000), no. 4, 395–430. MR 1808128, DOI 10.1007/s005260000046
Sigurd B. Angenent and Robertus van der Vorst, A priori bounds and renormalized Morse indices of solutions of an elliptic system, Ann. Inst. H. Poincaré C Anal. Non Linéaire 17 (2000), no. 3, 277–306 (English, with English and French summaries). MR 1771136, DOI 10.1016/S0294-1449(00)00110-4
Andrés I. Ávila and Jianfu Yang, On the existence and shape of least energy solutions for some elliptic systems, J. Differential Equations 191 (2003), no. 2, 348–376. MR 1978382, DOI 10.1016/S0022-0396(03)00017-2
A. Bahri and P.-L. Lions, Solutions of superlinear elliptic equations and their Morse indices, Comm. Pure Appl. Math. 45 (1992), no. 9, 1205–1215. MR 1177482, DOI 10.1002/cpa.3160450908
Jérôme Busca and Boyan Sirakov, Symmetry results for semilinear elliptic systems in the whole space, J. Differential Equations 163 (2000), no. 1, 41–56. MR 1755067, DOI 10.1006/jdeq.1999.3701
Ph. Clément, D. G. de Figueiredo, and E. Mitidieri, Positive solutions of semilinear elliptic systems, Comm. Partial Differential Equations 17 (1992), no. 5-6, 923–940. MR 1177298, DOI 10.1080/03605309208820869
D. G. de Figueiredo and P. L. Felmer, A Liouville-type theorem for elliptic systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 21 (1994), no. 3, 387–397. MR 1310633
Djairo G. de Figueiredo and Patricio L. Felmer, On superquadratic elliptic systems, Trans. Amer. Math. Soc. 343 (1994), no. 1, 99–116. MR 1214781, DOI 10.1090/S0002-9947-1994-1214781-2
Manuel Del Pino and Patricio L. Felmer, Spike-layered solutions of singularly perturbed elliptic problems in a degenerate setting, Indiana Univ. Math. J. 48 (1999), no. 3, 883–898. MR 1736974, DOI 10.1512/iumj.1999.48.1596
Josephus Hulshof and Robertus van der Vorst, Differential systems with strongly indefinite variational structure, J. Funct. Anal. 114 (1993), no. 1, 32–58. MR 1220982, DOI 10.1006/jfan.1993.1062
C.-S. Lin, W.-M. Ni, and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations 72 (1988), no. 1, 1–27. MR 929196, DOI 10.1016/0022-0396(88)90147-7
Enzo Mitidieri, A Rellich type identity and applications, Comm. Partial Differential Equations 18 (1993), no. 1-2, 125–151. MR 1211727, DOI 10.1080/03605309308820923
Wei-Ming Ni and Izumi Takagi, On the Neumann problem for some semilinear elliptic equations and systems of activator-inhibitor type, Trans. Amer. Math. Soc. 297 (1986), no. 1, 351–368. MR 849484, DOI 10.1090/S0002-9947-1986-0849484-2
Wei-Ming Ni and Izumi Takagi, On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math. 44 (1991), no. 7, 819–851. MR 1115095, DOI 10.1002/cpa.3160440705
Wei-Ming Ni and Izumi Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Math. J. 70 (1993), no. 2, 247–281. MR 1219814, DOI 10.1215/S0012-7094-93-07004-4
L. A. Peletier and R. C. A. M. Van der Vorst, Existence and nonexistence of positive solutions of nonlinear elliptic systems and the biharmonic equation, Differential Integral Equations 5 (1992), no. 4, 747–767. MR 1167492
Miguel Ramos, Susanna Terracini, and Christophe Troestler, Superlinear indefinite elliptic problems and Pohožaev type identities, J. Funct. Anal. 159 (1998), no. 2, 596–628. MR 1658097, DOI 10.1006/jfan.1998.3332
Boyan Sirakov, On the existence of solutions of Hamiltonian elliptic systems in $\mathbf R^N$, Adv. Differential Equations 5 (2000), no. 10-12, 1445–1464. MR 1785681
R. C. A. M. Van der Vorst, Variational identities and applications to differential systems, Arch. Rational Mech. Anal. 116 (1992), no. 4, 375–398. MR 1132768, DOI 10.1007/BF00375674