Spectral properties of -Laplacian problems with Neumann and mixed-type multi-point boundary conditions

✍ Scribed by Bryan P. Rynne

• Publisher: Elsevier Science
• Year: 2011
• Tongue: English
• Weight: 318 KB
• Volume: 74
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/j.na.2010.10.020

✦ Synopsis

We term the conditions (2) and (3), respectively, Neumann-type and Dirichlet-type boundary conditions, since they reduce to the standard Neumann and Dirichlet boundary conditions when α ± = 0. Given a suitable pair of boundary conditions, a number λ is an eigenvalue of the corresponding boundary value problem if there exists a non-trivial solution u (an eigenfunction). The spectrum of the problem is the set of eigenvalues. In this paper we obtain various spectral properties of these eigenvalue problems. We then use these properties to prove Rabinowitz-type, global bifurcation theorems for related bifurcation problems, and to obtain nonresonance conditions (in terms of the eigenvalues) for the solvability of related inhomogeneous problems.