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Title: The fibering method and its applications to nonlinear boundary value problem
Authors: Pohožaev, S. I.
Issue Date: 1999
Publisher: Università degli Studi di Trieste. Dipartimento di Scienze Matematiche
Source: S. I. Pohožaev (Pokhozhaev), "The fibering method and its applications to nonlinear boundary value problem", in: Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics, 31 (1999), pp. 235-305.
Series/Report no.: Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics
31 (1999)
We consider the fibering method, proposed in $\left[1\right]$ for
investigating some variational problems, and its applications to nonlinear
elliptic equations. Let X an d Y be Banach spaces, and let A be an
operator (nonlinear in general) acting from X to Y. We consider the
A\left(u\right)=h.\qquad\qquad\qquad\qquad & \left(1\right)\end{array}
The fibering method is based on representation of solutions of equation
$\left(1\right)$ in the form
Here t is a real parameter $\left(t\neq0\: in\: some\: open\: J\subseteq\mathbb{R}\right)$,
and v is a nonzero element of X satisfying the condition
Generally speaking, any functional satisfying a sufficiently general
condition can be taken as the functional H(t, v). ln particular, the
norm H(t, v)=$\parallel v\parallel$=1: in this case we get a so-called
spherical fibering; here a solution $u\neq0$ of (1) is sought in
the form (2), where $t\epsilon\mathbb{R}\backslash\left\{ 0\right\} $
and $v\epsilon S=\left\{ w\epsilon X:\parallel w\parallel=1\right\} $.
Thus, the essence of the fibering method consists in imbedding the
space X of the original problem (1) in the larger space $\mathbb{R}\times X$
and investigating the new problem of conditional solvability in the
space $\mathbb{R}\times X$ under condition (3). This method makes
it possible to get both new existence and nonexistence theorems for
solutions of nonlinear boundary value problems. Moreover, in the investigation
of solvability of boundary value problems this method makes it possible
to separate algebraic and topological factors of the problem, which
affect the number of solutions. The origin of this approach is in
the calculus of variations, where this fibration arises in a natural
way in investigating variational problems on relative extrema involving
a given normalizing parameter $t\neq0$. A description of the method
of spherical fibering and some of its applications are given in $\left[18\right],\left[19\right],\left[20\right],\left[21\right],\left[22\right]\left[12\right]$.
We remark that the elements of Otis method were used as far back as
in $\left[18\right]$ in establishing the \textquotedbl{}Fredholm
alternative\textquotedbl{} for nonlinear odd and homogeneous (mainly)
strongly closed operators.
Type: Article
ISSN: 0049-4704
Appears in Collections:Rendiconti dell'Istituto di Matematica dell'Università di Trieste: an International Journal of Mathematics vol.31 (1999)

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