The structure of a nonlinear elliptic operator

Abstract:

Consider the nonlinear Dirichlet problem $(1) - \Delta u - \lambda u + {u^3} = g$, for $u:\Omega \to \mathbb {R}$, $u|\partial \Omega = 0$, and $\Omega \subset {\mathbb {R}^n}$ connected and bounded, and let ${\lambda _i}$ be the $i$th eigenvalue of $- \Delta u$ on $\Omega$ with $u|\partial \Omega = 0$, $(i = 1,2, \ldots )$. 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:\[ \begin {array}{*{20}{c}} {H \times ( - \infty ,{\lambda _1} + \varepsilon )} & {\xrightarrow [ \approx ]{\alpha }} & {{\mathbb {R}^2} \times E} \\ {A \downarrow } & {} & { \downarrow G} \\ {H’ \times ( - \infty ,{\lambda _1} + \varepsilon )} & {\xrightarrow [ \approx ]{\beta }} & {{\mathbb {R}^2} \times E} \\ \end {array} \] 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., $SA$].