Homeomorphisms and Fredholm Theory for Perturbations of Nonlinear Fredholm Maps of Index Zero and of A-Proper Maps with Applications

2010 ◽  
pp. 339-363
Author(s):  
P. S. Milojević
Keyword(s):  
Fredholm Theory ◽  
Index Zero ◽  
Proper Maps ◽  
Fredholm Maps

scholarly journals GLOBAL BRANCH OF SOLUTIONS FOR NON-LINEAR SCHRÖDINGER EQUATIONS WITH DEEPENING POTENTIAL WELL

2006 ◽  
Vol 92 (3) ◽  
pp. 655-681 ◽  
Author(s):  
C. A. STUART ◽  
HUAN-SONG ZHOU
Keyword(s):  
Potential Well ◽  
Index Zero ◽  
Topological Methods ◽  
Open Set ◽  
Connected Sets ◽  
Non Linear ◽  
Asymptotic Linearization ◽  
Positive Eigenfunction ◽  
Fredholm Maps ◽  
Global Branch

We consider the stationary non-linear Schrödinger equation\begin{equation*}\Delta u + \{1 + \lambda g(x)\} u = f(u)\mbox{with}u \in H^{1} (\mathbb{R}^{N}), u \not\equiv 0,\end{equation*} where $\lambda >0$ and the functions $f$ and $g$ are such that\begin{equation*} \lim_{s \rightarrow 0}\frac{f(s)}{s} = 0 \mbox{and} 1 < \alpha + 1 = \lim _{|s| \rightarrow \infty}\frac{f(s)}{s} < \infty\end{equation*} and \begin{equation*} g(x)\equiv 0 \mbox{on} \bar{\Omega}, g(x)\in (0, 1] \mbox{on} {\mathbb{R}^{N}} \setminus {\overline{\Omega}} \mbox{and} \lim_{|x| \rightarrow + \infty} g(x) = 1 \end{equation*} for some bounded open set $\Omega \in \mathbb{R}^{N}$. We use topological methods to establish the existence of two connected sets $\mathcal{D}^{\pm}$ of positive/negative solutions in $\mathbb{R} \times W^{2, p} (\mathbb{R}^{N})$ where $p \in [2, \infty) \cap (\frac{N}{2},\infty)$ that cover the interval $(\alpha,\Lambda(\alpha))$ in the sense that \begin{align*} P \mathcal{D}^{\pm} & = (\alpha, \Lambda(\alpha)) \text{where}P(\lambda, u) = \lambda \text{and furthermore,} \\ \lim_{\lambda \rightarrow \Lambda(\alpha)-}\left\Vert u_{\lambda} \right\Vert _{L^{\infty} (\mathbb{R}^{N})} & = \lim_{\lambda \rightarrow \Lambda (\alpha )-} \left\Vert u_{\lambda} \right\Vert _{W^{2, p}(\mathbb{R}^{N})} = \infty \text{ for }(\lambda, u_{\lambda}) \in \mathcal{D}^{\pm}. \end{align*} The number $\Lambda(\alpha)$ is characterized as the unique value of $\lambda$ in the interval $(\alpha, \infty)$ for which the asymptotic linearization has a positive eigenfunction. Our work uses a degree for Fredholm maps of index zero.

On the product formula for the oriented degree for Fredholm maps of index zero between Banach manifolds

2002 ◽  
Vol 48 (6) ◽  
pp. 853-867 ◽  
Author(s):  
Pierluigi Benevieri ◽  
Massimo Furi ◽  
Maria Patrizia Pera
Keyword(s):  
Product Formula ◽  
Index Zero ◽  
Banach Manifolds ◽  
Fredholm Maps

scholarly journals Axiomatization of the degree of Fitzpatrick, Pejsachowicz and Rabier

2021 ◽  
Vol 24 (1) ◽  
Author(s):  
Julián López-Gómez ◽  
Juan Carlos Sampedro
Keyword(s):  
Uniqueness Theorems ◽  
Algebraic Multiplicity ◽  
Fredholm Operators ◽  
Index Zero ◽  
Crucial Step ◽  
Fredholm Maps

AbstractIn this paper, we prove an analogue of the uniqueness theorems of Führer [15] and Amann and Weiss [1] to cover the degree of Fredholm operators of index zero constructed by Fitzpatrick, Pejsachowicz and Rabier [13], whose range of applicability is substantially wider than for the most classical degrees of Brouwer [5] and Leray–Schauder [22]. A crucial step towards the axiomatization of this degree is provided by the generalized algebraic multiplicity of Esquinas and López-Gómez [8, 9, 25], $$\chi $$ χ , and the axiomatization theorem of Mora-Corral [28, 32]. The latest result facilitates the axiomatization of the parity of Fitzpatrick and Pejsachowicz [12], $$\sigma (\cdot ,[a,b])$$ σ ( · , [ a , b ] ) , which provides the key step for establishing the uniqueness of the degree for Fredholm maps.

scholarly journals HILBERT STRATIFOLDS AND A QUILLEN TYPE GEOMETRIC DESCRIPTION OF COHOMOLOGY FOR HILBERT MANIFOLDS

2018 ◽  
Vol 6 ◽  
Author(s):  
MATTHIAS KRECK ◽  
HAGGAI TENE
Keyword(s):  
Dimensional Space ◽  
Characteristic Classes ◽  
Geometric Description ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Singular Cohomology ◽  
Cobordism Theory ◽  
Proper Maps ◽  
Definition Of ◽  
Fredholm Maps

In this paper we use tools from differential topology to give a geometric description of cohomology for Hilbert manifolds. Our model is Quillen’s geometric description of cobordism groups for finite-dimensional smooth manifolds [Quillen, ‘Elementary proofs of some results of cobordism theory using steenrod operations’, Adv. Math., 7 (1971)]. Quillen stresses the fact that this construction allows the definition of Gysin maps for ‘oriented’ proper maps. For finite-dimensional manifolds one has a Gysin map in singular cohomology which is based on Poincaré duality, hence it is not clear how to extend it to infinite-dimensional manifolds. But perhaps one can overcome this difficulty by giving a Quillen type description of singular cohomology for Hilbert manifolds. This is what we do in this paper. Besides constructing a general Gysin map, one of our motivations was a geometric construction of equivariant cohomology, which even for a point is the cohomology of the infinite-dimensional space $BG$, which has a Hilbert manifold model. Besides that, we demonstrate the use of such a geometric description of cohomology by several other applications. We give a quick description of characteristic classes of a finite-dimensional vector bundle and apply it to a generalized Steenrod representation problem for Hilbert manifolds and define a notion of a degree of proper oriented Fredholm maps of index $0$.

scholarly journals Global Persistence of the Unit Eigenvectors of Perturbed Eigenvalue Problems in Hilbert Spaces: The Odd Multiplicity Case

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 561
Author(s):  
Pierluigi Benevieri ◽  
Alessandro Calamai ◽  
Massimo Furi ◽  
Maria Patrizia Pera
Keyword(s):  
Hilbert Spaces ◽  
Eigenvalue Problems ◽  
Eigenvalues And Eigenvectors ◽  
Index Zero ◽  
Global Continuation ◽  
Banach Manifolds ◽  
Unperturbed Problem ◽  
Fredholm Maps

We study the persistence of eigenvalues and eigenvectors of perturbed eigenvalue problems in Hilbert spaces. We assume that the unperturbed problem has a nontrivial kernel of odd dimension and we prove a Rabinowitz-type global continuation result. The approach is topological, based on a notion of degree for oriented Fredholm maps of index zero between real differentiable Banach manifolds.

Sign in / Sign up

Export Citation Format

Share Document