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The fibering method and its applications to nonlinear boundary value problem
Pohožaev, S. I.
1999
Abstract
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
equation
\[
\begin{array}{cc}
A\left(u\right)=h.\qquad\qquad\qquad\qquad & \left(1\right)\end{array}
\]
The fibering method is based on representation of solutions of equation
$\left(1\right)$ in the form
\[
u=tv.\qquad\qquad\qquad\qquad\qquad\left(2\right)
\]
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
\[
H\left(t,v\right)=c.\qquad\qquad\qquad
\]
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.
Series
Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics
31 (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.
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