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On global bifurcation for quasilinear elliptic systems on bounded domains. (English) Zbl 1165.35358
Summary: General second order quasilinear elliptic systems with nonlinear boundary conditions on bounded domains are formulated as nonlinear mappings between Sobolev spaces. It is shown that the linearized mapping is a Fredholm operator of index zero. This and the abstract global bifurcation theorem of J. Pejsachowicz and P. J. Rabier [J. Anal. Math. 76, 289–319 (1998; Zbl 0932.47046)] allow us to carry out bifurcation analysis directly on these elliptic systems. At the abstract level, we establish a unilateral global bifurcation result that is needed when studying positive solutions. Finally, we supply two examples of cross-diffusion population model and chemotaxis model to demonstrate how the theory can be applied.

35J55Systems of elliptic equations, boundary value problems (MSC2000)
35B32Bifurcation (PDE)
46T20Continuous and differentiable maps in nonlinear functional analysis
58C25Differentiable maps on manifolds (global analysis)
92D25Population dynamics (general)
92C17Cell movement (chemotaxis, etc.)
[1]Adams, Robert A.: Sobolev spaces, Pure appl. Math. 65 (1975) · Zbl 0314.46030
[2]Agmon, Shmuel: On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems, Comm. pure appl. Math. 15, 119-147 (1962) · Zbl 0109.32701 · doi:10.1002/cpa.3160150203
[3]Agmon, S.; Douglis, A.; Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. pure appl. Math. 12, 623-727 (1959) · Zbl 0093.10401 · doi:10.1002/cpa.3160120405
[4]Agmon, S.; Douglis, A.; Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II, Comm. pure appl. Math. 17, 35-92 (1964) · Zbl 0123.28706 · doi:10.1002/cpa.3160170104
[5]Agranovich, Mikhail; Denk, Robert; Faierman, Melvin: Weakly smooth nonselfadjoint spectral elliptic boundary problems, Math. top. 14, 138-199 (1997) · Zbl 0932.35158
[6]Agranovich, M. S.; Vishik, M. I.: Elliptic problems with a parameter and parabolic problems of general type, Uspekhi mat. Nauk 19, No. 3(117), 53-161 (1964) · Zbl 0137.29602
[7]Amann, Herbert: Dynamic theory of quasilinear parabolic equations. II. reaction – diffusion systems, Differential integral equations 3, No. 1, 13-75 (1990) · Zbl 0729.35062
[8]Blat, J.; Brown, K. J.: Global bifurcation of positive solutions in some systems of elliptic equations, SIAM J. Math. anal. 17, 1339-1353 (1986) · Zbl 0613.35008 · doi:10.1137/0517094
[9]Crandall, Michael G.; Rabinowitz, Paul H.: Bifurcation from simple eigenvalues, J. funct. Anal. 8, 321-340 (1971) · Zbl 0219.46015 · doi:10.1016/0022-1236(71)90015-2
[10]Dancer, E. N.: On positive solutions of some pairs of differential equations, II, J. differential equations 60, 236-258 (1985) · Zbl 0549.35024 · doi:10.1016/0022-0396(85)90115-9
[11]Dancer, E. N.: Bifurcation from simple eigenvalues and eigenvalues of geometric multiplicity one, Bull. lond. Math. soc. 34, 533-538 (2002) · Zbl 1027.58009 · doi:10.1112/S002460930200108X
[12]Denk, R.; Faierman, M.; Möller, M.: An elliptic boundary problem for a system involving a discontinuous weight, Manuscripta math. 108, No. 3, 289-317 (2002) · Zbl 1003.35049 · doi:10.1007/s002290200264
[13]Du, Yihong; Shi, Junping: Allee effect and bistability in a spatially heterogeneous predator – prey model, Trans. amer. Math. soc. 359, No. 9, 4557-4593 (2007) · Zbl 1189.35337 · doi:10.1090/S0002-9947-07-04262-6
[14]Du, Yihong; Shi, Junping: Some recent results on diffusive predator – prey models in spatially heterogeneous environment, Fields inst. Commun. 48, 95-135 (2006) · Zbl 1100.35041
[15]Fitzpatrick, P. M.; Pejsachowicz, Jacobo: Parity and generalized multiplicity, Trans. amer. Math. soc. 326, No. 1, 281-305 (1991) · Zbl 0754.47009 · doi:10.2307/2001865
[16]Fitzpatrick, P. M.; Pejsachowicz, Jacobo: Orientation and the Leray – Schauder theory for fully nonlinear elliptic boundary value problems, Mem. amer. Math. soc. 101, No. 483 (1993)
[17]Goldberg, Seymour: Unbounded linear operators. Theory and applications, (1985) · Zbl 0925.47001
[18]Gebran, Hicham G.; Stuart, Charles A.: Fredholm and properness properties of quasilinear elliptic systems of second order, Proc. edinb. Math. soc. (2) 48, No. 1, 91-124 (2005) · Zbl 1210.35091 · doi:10.1017/S0013091504000550
[19]Geymonat, G.; Grisvard, P.: Alcuni risultati di teoria spettrale per i problemi ai limiti lineari ellittici, Rend. sem. Mat. univ. Padova 38, 121-173 (1967) · Zbl 0182.18503 · doi:numdam:RSMUP_1967__38__121_0
[20]Krasnosel’skii, M. A.: Topological methods in the theory of nonlinear integral equations, (1964) · Zbl 0111.30303
[21]Kuto, Kousuke; Yamada, Yoshio: Multiple coexistence states for a prey – predator system with cross-diffusion, J. differential equations 197, No. 2, 315-348 (2004) · Zbl 1205.35116 · doi:10.1016/j.jde.2003.08.003
[22]Ladyzhenskaya, O. A.; Solonnikov, V. A.; Ural’tseva, N. N.: Linear and quasilinear equations of parabolic type, Transl. math. Monogr. 23 (1967) · Zbl 0174.15403
[23]Liu, Ping; Shi, Junping; Wang, Yuwen: Imperfect transcritical and pitchfork bifurcations, J. funct. Anal. 251, No. 2, 573-600 (2007) · Zbl 1139.47042 · doi:10.1016/j.jfa.2007.06.015
[24]López-Gómez, Julian: Spectral theory and nonlinear functional analysis, Chapman & Hall/CRC res. Notes math. 426 (2001) · Zbl 0978.47048
[25]Lou, Yuan; Ni, Wei-Ming: Diffusion, self-diffusion and cross-diffusion, J. differential equations 131, No. 1, 79-131 (1996) · Zbl 0867.35032 · doi:10.1006/jdeq.1996.0157
[26]Lou, Yuan; Ni, Wei-Ming: Diffusion vs cross-diffusion: an elliptic approach, J. differential equations 154, No. 1, 157-190 (1999) · Zbl 0934.35040 · doi:10.1006/jdeq.1998.3559
[27]Nakashima, Kimie; Yamada, Yoshio: Positive steady states for prey – predator models with cross-diffusion, Adv. differential equations 1, No. 6, 1099-1122 (1996) · Zbl 0863.35034
[28]Ni, Wei-Ming: Diffusion, cross-diffusion, and their spike-layer steady states, Notices amer. Math. soc. 45, No. 1, 9-18 (1998) · Zbl 0917.35047 · doi:http://www.ams.org/notices/199801/ni.pdf
[29]Ni, Wei-Ming: Qualitative properties of solutions to elliptic problems, Handb. differ. Equ., 157-233 (2004) · Zbl 1129.35401
[30]Pejsachowicz, Jacobo; Rabier, Patrick J.: Degree theory for C1 Fredholm mappings of index 0, J. anal. Math. 76, 289-319 (1998) · Zbl 0932.47046 · doi:10.1007/BF02786939
[31]Peetre, J.: Another approach to elliptic boundary problems, Comm. pure appl. Math. 14, 711-731 (1961) · Zbl 0104.07303 · doi:10.1002/cpa.3160140404
[32]Rabier, Patrick J.: Generalized Jordan chains and two bifurcation theorems of Krasnoselskii, Nonlinear anal. 13, No. 8, 903-934 (1989) · Zbl 0686.47047 · doi:10.1016/0362-546X(89)90021-7
[33]Rabier, Patrick J.: On the index and spectrum of differential operators on RN, Proc. amer. Math. soc. 135, No. 12, 3875-3885 (2007) · Zbl 1129.47034 · doi:10.1090/S0002-9939-07-08896-X
[34]Rabier, Patrick J.; Stuart, Charles A.: Fredholm and properness properties of quasilinear elliptic operators on RN, Math. nachr. 231, 129-168 (2001) · Zbl 0991.35021 · doi:10.1002/1522-2616(200111)231:1<129::AID-MANA129>3.0.CO;2-V
[35]Rabinowitz, Paul H.: Some global results for nonlinear eigenvalue problems, J. funct. Anal. 7, 487-513 (1971) · Zbl 0212.16504 · doi:10.1016/0022-1236(71)90030-9
[36]Restrepo, G.: Differentiable norms in Banach spaces, Bull. amer. Math. soc. 70, 413-414 (1964) · Zbl 0173.41304 · doi:10.1090/S0002-9904-1964-11121-6
[37]Shi, Junping: Persistence and bifurcation of degenerate solutions, J. funct. Anal. 169, No. 2, 494-531 (1999) · Zbl 0949.47050 · doi:10.1006/jfan.1999.3483
[38]Shigesada, N.; Kawasaki, K.; Teramoto, E.: Spatial segregation of interacting species, J. theoret. Biol. 79, 83-99 (1979)
[39]Wang, Xuefeng: Qualitative behavior of solutions of a chemotactic diffusion system: effects of motility and chemotaxis and dynamics, SIAM J. Math. anal. 31, 535-560 (2000) · Zbl 0990.92001 · doi:10.1137/S0036141098339897
[40]Wloka, J.: Partial differential equations, (1987)