This article is concerned with the bifurcation from infinity of the following elliptic system arising from biology
in a bounded domain . We regard this problem as a stationary problem of some reaction-diffusion system. By using a method of a pure dynamical nature, we will establish some multiplicity results on bifurcations from infinity for this system under an appropriate Landesman-Lazer type condition.
C.Castaing and M.Valadier, Convex Analysis and Measurable Multifunctions, Springer-Verlag, Berlin, 1977.
2.
X.Chang and Y.Li, Existence and multiplicity of nontrivial solutions for semilinear elliptic Dirichlet problems across resonance, Topol. Methods Nonlinear Anal.36 (2010), 285–310.
3.
R.Chiappinelli and D.G.de Figueiredo, Bifurcation from infinity and multiple solutions for an elliptic system, Differential Integral Equations6(4) (1993), 757–771. doi:10.57262/die/1370032232.
4.
R.Chiappinelli, J.Mawhin and R.Nugari, Bifurcation from infinity and multiple solutions for some Dirichlet problems with unbounded nonlinearities, Nonlinear Anal. TMA18 (1992), 1099–1112. doi:10.1016/0362-546X(92)90155-8.
5.
C.Conley, Isolated Invariant Sets and the Morse Index, Regional Conference Series in Mathematics, Vol. 38, Amer. Math. Soc., Providence RI, 1978.
6.
C.Conley and R.Easton, Isolated invariant sets and isolating blocks, Trans. Amer. Math. Soc.158 (1971), 35–61. doi:10.1090/S0002-9947-1971-0279830-1.
7.
D.G.de Figueiredo and E.Mitidieri, A maximum principle for an elliptic system and applications to semilinear problems, SIAM J. Math. Anal.17 (1986), 836–849. doi:10.1137/0517060.
8.
F.O.de Paiva and E.Massa, Semilinear elliptic problems near resonance with a nonprincipal eigenvalue, J. Math. Anal. Appl.342 (2008), 638–650. doi:10.1016/j.jmaa.2007.12.053.
9.
Y.Du and T.Ouyang, Bifurcation from infinity induced by a degeneracy in semilinear equations, Advanced Nonlinear Studies2(2) (2002), 117–132. doi:10.1515/ans-2002-0202.
10.
D.Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., Vol. 840, Springer-Verlag, Berlin, 1981.
11.
S.Koga and Y.Kuramoto, Localized patterns in reaction-diffusion systems, Progress of Theoretical Physics63(1) (1980), 106–121. doi:10.1143/PTP.63.106.
12.
A.C.Lazer and P.J.McKenna, On state solutions of a system of reaction-diffusion equations from biology, Nonlinear Anal., TMA6 (1982), 523–530. doi:10.1016/0362-546X(82)90045-1.
13.
C.Q.Li, D.S.Li and J.T.Wang, A remark on attractor bifurcation, Dynamics of PDE18(2) (2021), 157–172.
14.
C.Q.Li, D.S.Li and Z.J.Zhang, Dynamic bifurcation from infinity of nonlinear evolution equations, SIAM J. Appl. Dyn. Syst.16 (2017), 1831–1868. doi:10.1137/16M1107358.
15.
C.Q.Li and J.T.Wang, Bifurcation from infinity of the Schrödinger equation via invariant manifolds, Nonlinear Anal.213 (2021), 112490. doi:10.1016/j.na.2021.112490.
16.
D.S.Li, G.L.Shi and X.F.Song, Linking theorems of local semiflows on complete metric spaces, 2015, arXiv:1312.1868.
17.
D.S.Li and Z.Q.Wang, Local and global dynamic bifurcations of nonlinear evolution equations, Indiana Univ. Math. J.67(2) (2018), 583–621. doi:10.1512/iumj.2018.67.7292.
18.
Z.Liang and J.Su, Multiple solutions for semilinear elliptic boundary value problems with double resonance, J. Math. Anal. Appl.354 (2009), 147–158. doi:10.1016/j.jmaa.2008.12.053.
19.
T.Ma and S.H.Wang, Bifurcation Theory and Applications, World Sci. Ser. Nonlinear Sci. Ser. A, Vol. 53, World Scientific, Hackensack, NJ, 2005.
20.
J.Mawhin and K.Schmitt, Landesman-Lazer type problems at an eigenvalue of odd multiplicity, Results Math.14 (1988), 138–146. doi:10.1007/BF03323221.
21.
J.Mawhin and K.Schmitt, Nonlinear eigenvalue problems with the parameter near resonance, Ann. Polon. Math.51 (1990), 241–248. doi:10.4064/ap-51-1-241-248.
22.
C.McCord, Poincaré-Lefschetz duality for the homolopy Conley index, Trans. Amer. Math. Soc.329 (1992), 233–252.
23.
K.Mischaikow and M.Mrozek, Conley index, in: Handb. Dynam. Syst. 2, Elsevier, New York, 2002, pp. 393–460.
24.
P.Mottoni and F.Rothe, A singular perturbation analysis for a reaction-diffusion system describing pattern formation, Stud. Appl. Math.63(3) (1980), 227–247. doi:10.1002/sapm1980633227.
25.
M.Mrozek and R.Srzednicki, On time-duality of the Conley index, Results Math.24 (1993), 161–167. doi:10.1007/BF03322325.
26.
N.S.Papageorgiou and F.Papalini, Multiple solutions for nearly resonant nonlinear Dirichlet problems, Potential Anal.37 (2012), 247–279. doi:10.1007/s11118-011-9255-8.
27.
P.H.Rabinowitz, On bifurcation from infinity, J. Differential Equations14 (1973), 462–475. doi:10.1016/0022-0396(73)90061-2.
28.
F.Rothe, Some analytical results about a simple reaction-diffusion system for morphogenesis, J. Math. Biol.7(4) (1979), 375–384. doi:10.1007/BF00275155.
29.
F.Rothe, Global existence of branches of stationary solutions for a system of reaction-diffusion equations from biology, Nonlinear Anal., TMA5 (1981), 487–798. doi:10.1016/0362-546X(81)90097-3.
30.
F.Rothe and P.de Morttoni, A simple system of reaction-diffusion equations describing morphogenesis: Asymptotic behavior, Annali di Matematica Pura ed Applicata122 (1979), 141–157. doi:10.1007/BF02411692.
31.
K.P.Rybakowski, The Homotopy Index and Partial Differential Equations, Springer-Verlag, Berlin, 1987.
32.
J.C.Saut and R.Temam, Generic properties of nonlinear boundary value problems, Comm. Partial Differential Equations4 (1979), 293–319. doi:10.1080/03605307908820096.
33.
K.Schmitt and Z.Q.Wang, On bifurcation from infinity for potential operators, Differential Integral Equations4 (1991), 933–943.
34.
J.Su, Existence and multiplicity results for classes of elliptic resonant problems, J. Math. Anal. Appl.273 (2002), 565–579. doi:10.1016/S0022-247X(02)00274-3.
35.
K.Teng and X.Wu, Multiplicity results for semilinear resonant elliptic problems with discontinuous nonlinearities, Nonlinear Anal. TMA68(6) (2008), 1652–1667. doi:10.1016/j.na.2006.12.050.
36.
J.T.Wang, D.S.Li and J.Q.Duan, On the shape Conley index theory of semiflows on complete metric spaces, Discrete Contin. Dyn. Syst. – A36(3) (2016), 1629–1647. doi:10.3934/dcds.2016.36.1629.
37.
J.R.WardJr., A global continuation theorem and bifurcation from infinity for infinite-dimensional dynamical systems, Proc. Roy. Soc. Edinburgh Sect. A126 (1996), 725–738. doi:10.1017/S0308210500023039.
38.
L.Zhou and D.S.Li, Global dynamic bifurcation of local semiflows and nonlinear evolution equations, J. Differential Equations300 (2021), 625–659. doi:10.1016/j.jde.2021.07.035.
