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A note on an equation with critical exponent. (English) Zbl 0646.35027
We first improve slightly results in the author’s earlier work [J. Differ. Equations (to appear)] and use the result to answer a question of Brezis. More precisely, we assume that f has polynomial growth, that \(u_ o\in \dot W^{1,2}(\Omega _ 0)\cap L^{\infty}(\Omega _ 0)\) that if \(-\Delta u_ 0=f(u_ 0)\) in \(\Omega _ 0\) and that the corresponding linearized equation at \(u_ 0\) has only the trivial solution. We prove that, if \(\Omega _ n\) is near \(\Omega _ 0\) in a rather weak sense then there is a unique solution \(u_ n\in \dot W^{1,2}(\Omega _ n)\cap L^{\infty}(\Omega _ n)\) of \(-\Delta u=f(u)\) in \(\Omega _ n\) near \(u_ 0\) (in a suitable L r norm).
We then combine this with results of A. Bahri and J.-M. Coron [Commun. Pure Appl. Math. 41, 253-294 (1988)] and J. C. Saut and R. Temam [Commun. Partial Differ. Equation 4, 293-319 (1979; Zbl 0462.35016)] to construct, for each \(m\geq 2\), a domain \(\Omega\) diffeomorphic to a ball in R m for which the equation \(-\Delta u=u^{(m+2)(m-2)^{-1}}\) in \(\Omega\), \(u=0\) on \(\partial \Omega\) has a nontrivial positive solution. This answers a question of H. Brezis [Commun. Pure Appl. Math. 39, Suppl., S 17-S 39 (1986; Zbl 0612.35052)].
Moreover, these solutions persist if we replace \(u^{(m+2)(m-2)^{-1}}\) by u p where p is near \((m+2)(m-2)^{-1}\). This partially answers another question in Brezis’s paper [op. cit.].
Reviewer: E.N.Dancer

35J60 Nonlinear elliptic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
58E99 Variational problems in infinite-dimensional spaces
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