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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)
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.
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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