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The structure of a nonlinear elliptic operator


Authors: P. T. Church, E. N. Dancer and J. G. Timourian
Journal: Trans. Amer. Math. Soc. 338 (1993), 1-42
MSC: Primary 35J65; Secondary 47H15, 47N20, 58C27
DOI: https://doi.org/10.1090/S0002-9947-1993-1124165-2
MathSciNet review: 1124165
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Abstract: Consider the nonlinear Dirichlet problem $ (1) - \Delta u - \lambda u + {u^3} = g$, for $ u:\Omega \to \mathbb{R}$, $ u\vert\partial \Omega = 0$, and $ \Omega \subset {\mathbb{R}^n}$ connected and bounded, and let $ {\lambda _i}$ be the $ i$th eigenvalue of $ - \Delta u$ on $ \Omega $ with $ u\vert\partial \Omega = 0$, $ (i = 1,2, \ldots )$. Define a map $ {A_\lambda }:H \to H\prime $ by $ {A_\lambda }(u) = - \Delta u - \lambda u + {u^3}$, for either the Sobolev space $ W_0^{1,2}(\Omega ) = H = H\prime$ (if $ n \leq 4)$ or the Hölder spaces $ C_0^{2,\alpha }(\bar \Omega ) = H$ and $ {C^{0,\alpha }}(\bar \Omega ) = H\prime $ (if $ \partial \Omega $ is $ {C^{2,\alpha }}$ ), and define $ A:H \times \mathbb{R} \to H\prime \times \mathbb{R}$ by $ A(u,\lambda ) = ({A_\lambda }(u),\lambda )$. Let $ G:{\mathbb{R}^2} \times E \to {\mathbb{R}^2} \times E$ be the global cusp map given by $ G(s,t,v) = ({s^3} - ts,t,v)$, and let $ F:\mathbb{R} \times E \to \mathbb{R} \times E$ be the global fold map given by $ F(t,v) = ({t^2},v)$, where $ E$ is any Fréchet space.

Theorem 1. If $ H = H\prime = W_0^{1,2}(\Omega )$, assume in addition that $ n \leqslant 3$. There exit $ \varepsilon > 0$ and homeomorphisms $ \alpha $ and $ \beta $ such that the following diagram commutes:

\begin{displaymath}\begin{array}{*{20}{c}} {H \times ( - \infty ,{\lambda _1} + ... ... \approx ]{\beta }} & {{\mathbb{R}^2} \times E} \\ \end{array} \end{displaymath}

The analog for $ {A_\lambda }$ with $ {\lambda _1} < \lambda < {\lambda _1} + \varepsilon $ is also given. In a very strong sense this theorem is a perturbation result for the problem (1): As $ g$ (and $ \lambda $) are perturbed, it shows how the number of solutions $ u$ of (1) varies; in particular, that number is always $ 1$, $ 2$ or $ 3$ for $ \lambda < {\lambda _1} + \varepsilon $.

A point $ u \in H$ is a fold point of $ A$ if the germ of $ A$ at $ u$ is $ {C^0}$ equivalent to the germ of $ F$ at $ (0,0)$ (i.e. under homeomorphic coordinate changes in domain near $ u$ and in range near $ A(u)$, $ A$ becomes $ F$), and the singular set $ SA$ is the set of points at which $ A$ fails to be a local diffeomorphism. For larger values of $ \lambda $ our information is limited:

Theorem 2. Consider the Sobolev case with $ n \leqslant 4$ and $ \partial \Omega \,{C^\infty }$. For all $ \lambda \in \mathbb{R}$, (i) $ \operatorname{int} (SA) = \emptyset $; (ii) there is a dense subset $ \Gamma $ in $ SA$ of fold points, and (iii) for $ \lambda < {\lambda _2}$, $ SA$ [resp., for $ n \leqslant 3$ and $ \lambda < {\lambda _2}$, $ SA - \Gamma $] is a real analytic submanifold of codimension $ 1$ in $ H \times \mathbb{R}$ [resp., $ SA$].


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DOI: https://doi.org/10.1090/S0002-9947-1993-1124165-2
Keywords: Nonlinear partial differential equations, elliptic boundary value problem, nonlinear Dirichlet problem, singularity theory in infinite dimensions, fold map, cusp map
Article copyright: © Copyright 1993 American Mathematical Society

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