On the generalized spectrum for second-order elliptic systems

Cantrell, Robert Stephen; Cosner, Chris

doi:10.1090/S0002-9947-1987-0896026-2

On the generalized spectrum for second-order elliptic systems
HTML articles powered by AMS MathViewer

by Robert Stephen Cantrell and Chris Cosner PDF

Trans. Amer. Math. Soc. 303 (1987), 345-363 Request permission

Abstract:

We consider the system of homogeneous Dirichlet boundary value problems $({\ast })$ \[ {L_1}u = \lambda [{a_{11}}(x)u + {a_{12}}(x)v],\quad {L_2}v = \mu [{a_{12}}(x)u + {a_{22}}(x)v]\] in a smooth bounded domain $\Omega \subseteq {{\mathbf {R}}^N}$, where ${L_1}$ and ${L_2}$ are formally self-adjoint second-order strongly uniformly elliptic operators. Using linear perturbation theory, continuation methods, and the Courant-Hilbert variational eigenvalue characterization, we give a detailed qualitative and quantitative description of the real generalized spectrum of $({\ast })$, i.e., the set $(\lambda , \mu ) \in {{\mathbf {R}}^2}: ({\ast })$ has a nontrivial solution. The generalized spectrum, a term introduced by Protter in 1979, is of considerable interest in the theory of linear partial differential equations and also in bifurcation theory, as it is the set of potential bifurcation points for associated semilinear systems.

References

J. C. Alexander and Stuart S. Antman, Global and local behavior of bifurcating multidimensional continua of solutions for multiparameter nonlinear eigenvalue problems, Arch. Rational Mech. Anal. 76 (1981), no. 4, 339–354. MR 628173, DOI 10.1007/BF00249970
J. C. Alexander and Stuart S. Antman, Global behavior of solutions of nonlinear equations depending on infinite-dimensional parameters, Indiana Univ. Math. J. 32 (1983), no. 1, 39–62. MR 684754, DOI 10.1512/iumj.1983.32.32004
K. J. Brown and J. C. Eilbeck, Bifurcation, stability diagrams, and varying diffusion coefficients in reaction-diffusion equations, Bull. Math. Biol. 44 (1982), no. 1, 87–102. MR 665619, DOI 10.1016/S0092-8240(82)80033-5
Robert Stephen Cantrell, A homogeneity condition guaranteeing bifurcation in multiparameter nonlinear eigenvalue problems, Nonlinear Anal. 8 (1984), no. 2, 159–169. MR 734449, DOI 10.1016/0362-546X(84)90067-1
Robert Stephen Cantrell, On coupled multiparameter nonlinear elliptic systems, Trans. Amer. Math. Soc. 294 (1986), no. 1, 263–285. MR 819947, DOI 10.1090/S0002-9947-1986-0819947-4
Robert Stephen Cantrell and Chris Cosner, On the steady-state problem for the Volterra-Lotka competition model with diffusion, Houston J. Math. 13 (1987), no. 3, 337–352. MR 916141

On the uniqueness of positive solutions in the Lotka-Volterra competition model with diffusion

Robert Stephen Cantrell and Klaus Schmitt, On the eigenvalue problem for coupled elliptic systems, SIAM J. Math. Anal. 17 (1986), no. 4, 850–862. MR 846393, DOI 10.1137/0517061
R. Courant and D. Hilbert, Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, N.Y., 1953. MR 0065391
P. M. Fitzpatrick, I. Massabò, and J. Pejsachowicz, Complementing maps, continuation and global bifurcation, Bull. Amer. Math. Soc. (N.S.) 9 (1983), no. 1, 79–81. MR 699319, DOI 10.1090/S0273-0979-1983-15161-3
P. M. Fitzpatrick, I. Massabò, and J. Pejsachowicz, On the covering dimension of the set of solutions of some nonlinear equations, Trans. Amer. Math. Soc. 296 (1986), no. 2, 777–798. MR 846606, DOI 10.1090/S0002-9947-1986-0846606-4

Topics in nonlinear analysis

J. Ize, I. Massabò, J. Pejsachowicz, and A. Vignoli, Structure and dimension of global branches of solutions to multiparameter nonlinear equations, Trans. Amer. Math. Soc. 291 (1985), no. 2, 383–435. MR 800246, DOI 10.1090/S0002-9947-1985-0800246-0
M. H. Protter, The generalized spectrum of second-order elliptic systems, Rocky Mountain J. Math. 9 (1979), no. 3, 503–518. MR 528748, DOI 10.1216/RMJ-1979-9-3-503
Franz Rellich, Perturbation theory of eigenvalue problems, Gordon and Breach Science Publishers, New York-London-Paris, 1969. Assisted by J. Berkowitz; With a preface by Jacob T. Schwartz. MR 0240668
J. T. Schwartz, Nonlinear functional analysis, Notes on Mathematics and its Applications, Gordon and Breach Science Publishers, New York-London-Paris, 1969. Notes by H. Fattorini, R. Nirenberg and H. Porta, with an additional chapter by Hermann Karcher. MR 0433481
K. Uhlenbeck, Generic properties of eigenfunctions, Amer. J. Math. 98 (1976), no. 4, 1059–1078. MR 464332, DOI 10.2307/2374041