Please use this identifier to cite or link to this item: http://hdl.handle.net/10077/4328
 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 Mathematics31 (1999) Abstract: We consider the fibering method, proposed in $\left[1\right]$ forinvestigating some variational problems, and its applications to nonlinearelliptic equations. Let X an d Y be Banach spaces, and let A be anoperator (nonlinear in general) acting from X to Y. We consider theequation $\begin{array}{cc}A\left(u\right)=h.\qquad\qquad\qquad\qquad & \left(1\right)\end{array}$ 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 generalcondition can be taken as the functional H(t, v). ln particular, thenorm H(t, v)=$\parallel v\parallel$=1: in this case we get a so-calledspherical fibering; here a solution $u\neq0$ of (1) is sought inthe 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 thespace X of the original problem (1) in the larger space $\mathbb{R}\times X$and investigating the new problem of conditional solvability in thespace $\mathbb{R}\times X$ under condition (3). This method makesit possible to get both new existence and nonexistence theorems forsolutions of nonlinear boundary value problems. Moreover, in the investigationof solvability of boundary value problems this method makes it possibleto separate algebraic and topological factors of the problem, whichaffect the number of solutions. The origin of this approach is inthe calculus of variations, where this fibration arises in a naturalway in investigating variational problems on relative extrema involvinga given normalizing parameter $t\neq0$. A description of the methodof 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 asin $\left[18\right]$ in establishing the \textquotedbl{}Fredholmalternative\textquotedbl{} for nonlinear odd and homogeneous (mainly)strongly closed operators. Type: Article URI: http://hdl.handle.net/10077/4328 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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