Ukrainian Mathematical Bulletin

Volume 2, Issue 1, 2005 pp. 39-52.

Bounded Components of Positive Solutions of Nonlinear Abstract Equations

Authors: Santiago Cano-Casanova, Julian Lopez-Gomez and Marcela Molina-Meyer
Author institution: Santiago Cano-Casanova, Departamento de Matematica Aplicada y Computacion Universidad Pontificia Comillas de Madrid, 28015-Madrid, Spain Julian Lopez-Gomez, Departamento de Matematica Aplicada Universidad Complutense de Madrid, 28040-Madrid, Spain Marcela Molina-Meyer, Departamento de Matematicas Universidad Carlos III de Madrid, 28911-Leganes, Madrid, Spain

Summary: In this work a general class of nonlinear abstract equations satisfying a {\it generalized strong maximum principle} is considered in order to show that any bounded component of positive solutions bifurcating from the curve of trivial states $(\lambda,u)=(\lambda,0)$ at a nonlinear eigenvalue $\lambda=\lambda_0$ must meet the curve of trivial states $(\lambda,0)$ at another singular value $\lambda_1\neq \lambda_0$. Since the unilateral theorems of P. H. Rabinowitz \cite[Theorems 1.27 and 1.40]{Ra71} are not true as originally stated (c.f. the counterexample of E. N. Dancer \cite{Da}), in order to get our main result the unilateral theorem of J. L\'{o}pez-G\'{o}mez \cite[Theorem 6.4.3]{Lo01} is required.

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