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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

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