The paper is concerned with the local and global bifurcation structure of positive solutions ${u,v\in H^1_0(\Omega)}$ of the system

$$\left\{\begin{array}{lll}-\Delta u + u = \mu_1u^3+\beta v^2u &\quad\text{in $\Omega$}\\-\Delta v + v = \mu_2v^3+\beta u^2v &\quad\text{in $\Omega$}\nonumber\end{array}\right.$$

of nonlinear Schrodinger (or Gross-Pitaevskii) type equations in ${\Omega\subset\mathbb {R}^N}$, N ≤ 3. The system arises in nonlinear optics and in the Hartree–Fock theory for a double condensate. Local and global bifurcations in terms of the nonlinear coupling parameter β of the system are investigated by using spectral analysis and by establishing a new Liouville type theorem for nonlinear elliptic systems which provides a-priori bounds of solution branches. If the domain is radial, possibly unbounded, then we also control the nodal structure of a certain weighted difference of the components of the solutions along the bifurcating branches.

/pdf/a-liouville-theorem-a-priori-bounds-and-bifurcating-branches-8lbqjjdejp.pdf

A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system

On determine toutes les relations universelles qui existent entre les indices de point fixe {I(f n )} n=1,2,... des iterees f n d'une application f:V→Y ou Y est un ENR, VCY est un sous-ensemble ouvert

https://link.springer.com/content/pdf/10.1007/BF01394243.pdf

Fixed point indices of iterated maps

Periodic solutions of some nonlinear, autonomous functional differential equations. II

The fixed point index and fixed point theorems

Foreword Preface to the Second Edition Preface 1. Maximization, Minimization, and Motivation 2. Vectors and Matrices 3. Diagonalization and Canonical Forms for Symmetric Matrices 4. Reduction of General Symmetric Matrices to Diagonal Form 5. Constrained Maxima 6. Functions of Matrices 7. Variational Description of Characteristic Roots 8. Inequalities 9. Dynamic Programming 10. Matrices and Differential Equations 11. Explicit Solutions and Canonical Forms 12. Symmetric Function, Kronecker Products and Circulants 13. Stability Theory 14. Markoff Matrices and Probability Theory 15. Stochastic Matrices 16. Positive Matrices, Perron's Theorem, and Mathematical Economics 17. Control Processes 18. Invariant Imbedding 19. Numerical Inversion of the Laplace Transform and Tychonov Regularization Appendix A. Linear Equations and Rank Appendix B. The Quadratic Form of Selberg Appendix C. A Method of Hermite Appendix D. Moments and Quadratic Forms Index.

Introduction to Matrix Analysis. By Richard Bellman. 2nd Edition. Pp. xxiii, 403. £7·50. (McGraw-Hill.)

Applied Equivariant Degree

The aim of this survey is to give a profound introduction to equivariant degree theory, avoiding as far as possible technical details and highly theoretical background. We describe the equivariant degree and its relation to the Brouwer degree for several classes of symmetry groups, including also the equivariant gradient degree, and particularly emphasizing the algebraic, analytical, and topological tools for its effective calculation, the latter being illustrated by six concrete examples. The paper concludes with a brief sketch of the construction and interpretation of the equivariant degree.

A short treatise on the equivariant degree theory and its applications

In an earlier paper we generalized the notion of a hyperbolic set and proved that the Shadowing Lemma remains valid, for C1-maps which need not be invertible. Here we establish the existence of (generalized) hyperbolic structures along transversal homoclinic trajectories of C1-maps. The hyperbolic structure and shadowing are then used to give a new proof of a result due to Hale and Lin (and silnikov) on symbolic dynamics forall trajectories sufficiently close to a transversal homoclinic trajectory. The result is applied to a Poincare map without continuous inverse, which is associated with a periodic orbit of an autonomous differential delay equation.

Hyperbolic sets, transversal homoclinic trajectories, and symbolic dynamics for C1-maps in banach spaces

Ein Satz über den Leray-Schauderschen Abbildungsgrad

The main result of this paper is the following theorem on the generalized fixed point index for locally condensing mappings: Let X be a subset of a Banach space such that there exists a locally finite covering {Ci|i∈I} of X by closed, convex subsets of X. If s=pt with p prime and t∈N, M an open subset of X and g: D[g]→X with D[g]⊂X and M⊂D[gS] such that g|M and gS|M are locally condensing and the fixed point set F of gS|M is compact and g(F)⊂M, then iX(gS,M)≡iX(g,M) mod p. This theorem can be applied in the theory of asymptotic fixed point theorems: An example may be found at the end of this paper.

Heinrich Steinlein

Papers

Applied Equivariant Degree

A short treatise on the equivariant degree theory and its applications

Hyperbolic sets, transversal homoclinic trajectories, and symbolic dynamics for C1-maps in banach spaces

Ein Satz über den Leray-Schauderschen Abbildungsgrad

Über die Verallgemeinerten Fixpunktindizes von Iterierten Verdichtender Abbildungen