Global structure of positive solution sets of nonlinear operator equations. (English) Zbl 1246.47013
Existence of unbounded maximal continua of positive solutions bifurcating from infinity for a class of nonlinear operator equations in the framework of Banach spaces ordered by a positive cone is studied in this article. The main salient feature in this paper is that, contrary to well-known results in this direction, the nonlinear terms involved are not asymptotically linear. The method of proof differs from the usual ones in this domain: first a sequence of unbounded subcontinua in a pipe (a notion which is introduced in the paper) bifurcating from nontrivial solutions is obtained by using topological degree, and then a limiting process (superior limit) in metric spaces gives the result. This argument leads to the main result (Theorem 3.1), and then a long list of variants (Theorems 3.2 to 3.10) proving existence of at least an asymptotic bifurcation point is given. The fixed point index is also an important tool for the proofs. An application is given to a nonlinear elliptic problem on a domain which is the complement of a disk in the space \(\mathbb{R}^n\). It is allowed in these results that the nonlinear terms may be non-positone.
Reviewer: Jesús Hernández (Madrid)
MSC:
| 47H07 | Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces |
| 47H10 | Fixed-point theorems |
| 58E07 | Variational problems in abstract bifurcation theory in infinite-dimensional spaces |
| 47H11 | Degree theory for nonlinear operators |
| 47J10 | Nonlinear spectral theory, nonlinear eigenvalue problems |
| 47J15 | Abstract bifurcation theory involving nonlinear operators |
| 35J65 | Nonlinear boundary value problems for linear elliptic equations |
Keywords:
asymptotic bifurcation; positive solution; fixed point index; topological degree; nonlinear eigenvalue problems; global bifurcationReferences:
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