Ground state and non-ground state solutions of some strongly coupled elliptic systems

Bonheure, Denis; Moreira dos Santos, Ederson; Ramos, Miguel

doi:10.1090/S0002-9947-2011-05452-8

Ground state and non-ground state solutions of some strongly coupled elliptic systems
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by Denis Bonheure, Ederson Moreira dos Santos and Miguel Ramos PDF

Trans. Amer. Math. Soc. 364 (2012), 447-491 Request permission

Abstract:

We study an elliptic system of the form $Lu = \left | v\right |^{p-1} v$ and $Lv=\left | u\right |^{q-1} u$ in $\Omega$ with homogeneous Dirichlet boundary condition, where $Lu:=-\Delta u$ in the case of a bounded domain and $Lu:=-\Delta u + u$ in the cases of an exterior domain or the whole space $\mathbb {R}^N$. We analyze the existence, uniqueness, sign and radial symmetry of ground state solutions and also look for sign changing solutions of the system. More general non-linearities are also considered.

References

Claudianor Oliveira Alves, Paulo César Carrião, and Olímpio Hiroshi Miyagaki, On the existence of positive solutions of a perturbed Hamiltonian system in ${\Bbb R}^N$, J. Math. Anal. Appl. 276 (2002), no. 2, 673–690. MR 1944783, DOI 10.1016/S0022-247X(02)00413-4
Angelo Alvino, Guido Trombetti, Pierre-Louis Lions, and Silvano Matarasso, Comparison results for solutions of elliptic problems via symmetrization, Ann. Inst. H. Poincaré C Anal. Non Linéaire 16 (1999), no. 2, 167–188 (English, with English and French summaries). MR 1674768, DOI 10.1016/S0294-1449(99)80011-0
A. Alvino, P.-L. Lions, and G. Trombetti, A remark on comparison results via symmetrization, Proc. Roy. Soc. Edinburgh Sect. A 102 (1986), no. 1-2, 37–48. MR 837159, DOI 10.1017/S0308210500014475
Vieri Benci and Giovanna Cerami, Positive solutions of some nonlinear elliptic problems in exterior domains, Arch. Rational Mech. Anal. 99 (1987), no. 4, 283–300. MR 898712, DOI 10.1007/BF00282048
Denis Bonheure and Miguel Ramos, Multiple critical points of perturbed symmetric strongly indefinite functionals, Ann. Inst. H. Poincaré C Anal. Non Linéaire 26 (2009), no. 2, 675–688 (English, with English and French summaries). MR 2504048, DOI 10.1016/j.anihpc.2008.06.002
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
Giovanna Cerami and Mónica Clapp, Sign changing solutions of semilinear elliptic problems in exterior domains, Calc. Var. Partial Differential Equations 30 (2007), no. 3, 353–367. MR 2332418, DOI 10.1007/s00526-007-0092-y
Giuseppe Chiti, Rearrangements of functions and convergence in Orlicz spaces, Applicable Anal. 9 (1979), no. 1, 23–27. MR 536688, DOI 10.1080/00036817908839248
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
Ph. Clément and E. Mitidieri, On a class of nonlinear elliptic systems, Sūrikaisekikenkyūsho K\B{o}kyūroku 1009 (1997), 132–140. Nonlinear evolution equations and their applications (Japanese) (Kyoto, 1996). MR 1630387
Philippe Clément, Patricio Felmer, and Enzo Mitidieri, Homoclinic orbits for a class of infinite-dimensional Hamiltonian systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 24 (1997), no. 2, 367–393. MR 1487960
Ph. Clément and R. C. A. M. Van der Vorst, On a semilinear elliptic system, Differential Integral Equations 8 (1995), no. 6, 1317–1329. MR 1329843
Robert Dalmasso, Existence and uniqueness of positive solutions of semilinear elliptic systems, Nonlinear Anal. 39 (2000), no. 5, Ser. A: Theory Methods, 559–568. MR 1727272, DOI 10.1016/S0362-546X(98)00221-1
Robert Dalmasso, Existence and uniqueness of positive radial solutions for the Lane-Emden system, Nonlinear Anal. 57 (2004), no. 3, 341–348. MR 2064095, DOI 10.1016/j.na.2004.02.018
Djairo G. de Figueiredo, Semilinear elliptic systems: existence, multiplicity, symmetry of solutions, Handbook of differential equations: stationary partial differential equations. Vol. V, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2008, pp. 1–48. MR 2497896, DOI 10.1016/S1874-5733(08)80008-3
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
Djairo G. De Figueiredo and Jianfu Yang, Decay, symmetry and existence of solutions of semilinear elliptic systems, Nonlinear Anal. 33 (1998), no. 3, 211–234. MR 1617988, DOI 10.1016/S0362-546X(97)00548-8
João Marcos do Ó and Pedro Ubilla, A multiplicity result for a class of superquadratic Hamiltonian systems, Electron. J. Differential Equations (2003), No. 15, 14. MR 1958050
E. M. dos Santos, Multiplicity of solutions for a fourth-order quasilinear nonhomogeneous equation, J. Math. Anal. Appl. 342 (2008), no. 1, 277–297. MR 2440796, DOI 10.1016/j.jmaa.2007.11.056
Alberto Ferrero, Filippo Gazzola, and Tobias Weth, Positivity, symmetry and uniqueness for minimizers of second-order Sobolev inequalities, Ann. Mat. Pura Appl. (4) 186 (2007), no. 4, 565–578. MR 2317778, DOI 10.1007/s10231-006-0019-9
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften, Vol. 224, Springer-Verlag, Berlin-New York, 1977. MR 0473443, DOI 10.1007/978-3-642-96379-7
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
Josephus Hulshof and Robertus C. A. M. van der Vorst, Asymptotic behaviour of ground states, Proc. Amer. Math. Soc. 124 (1996), no. 8, 2423–2431. MR 1363170, DOI 10.1090/S0002-9939-96-03669-6
P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoamericana 1 (1985), no. 1, 145–201. MR 834360, DOI 10.4171/RMI/6
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
Enzo Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $\textbf {R}^N$, Differential Integral Equations 9 (1996), no. 3, 465–479. MR 1371702
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
Angela Pistoia and Miguel Ramos, Locating the peaks of the least energy solutions to an elliptic system with Dirichlet boundary conditions, NoDEA Nonlinear Differential Equations Appl. 15 (2008), no. 1-2, 1–23. MR 2408342, DOI 10.1007/s00030-007-4066-8
Miguel Ramos and Sérgio H. M. Soares, On the concentration of solutions of singularly perturbed Hamiltonian systems in $\Bbb R^N$, Port. Math. (N.S.) 63 (2006), no. 2, 157–171. MR 2229874
Miguel Ramos and Hugo Tavares, Solutions with multiple spike patterns for an elliptic system, Calc. Var. Partial Differential Equations 31 (2008), no. 1, 1–25. MR 2342612, DOI 10.1007/s00526-007-0103-z
M. Ramos, H. Tavares, and W. Zou, A Bahri-Lions theorem revisited, Adv. Math. 222 (2009), no. 6, 2173–2195. MR 2562780, DOI 10.1016/j.aim.2009.07.013
Bernhard Ruf, Superlinear elliptic equations and systems, Handbook of differential equations: stationary partial differential equations. Vol. V, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2008, pp. 211–276. MR 2497908, DOI 10.1016/S1874-5733(08)80010-1
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
Giorgio Talenti, Elliptic equations and rearrangements, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 3 (1976), no. 4, 697–718. MR 601601
William C. Troy, Symmetry properties in systems of semilinear elliptic equations, J. Differential Equations 42 (1981), no. 3, 400–413. MR 639230, DOI 10.1016/0022-0396(81)90113-3
R. C. A. M. Van der Vorst, Best constant for the embedding of the space $H^2\cap H^1_0(\Omega )$ into $L^{2N/(N-4)}(\Omega )$, Differential Integral Equations 6 (1993), no. 2, 259–276. MR 1195382
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
Xu Jia Wang, Sharp constant in a Sobolev inequality, Nonlinear Anal. 20 (1993), no. 3, 261–268. MR 1202203, DOI 10.1016/0362-546X(93)90162-L
Michel Willem, Minimax theorems, Progress in Nonlinear Differential Equations and their Applications, vol. 24, Birkhäuser Boston, Inc., Boston, MA, 1996. MR 1400007, DOI 10.1007/978-1-4612-4146-1
Michel Willem, Principes d’analyse fonctionnelle, Nouvelle Bibliothèque Mathématique [New Mathematics Library], vol. 9, Cassini, Paris, 2007 (French). MR 2567317