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Critical Sobolev exponent problem in $${\mathbb{R}{}}^ n(n{\geq{}}4)$$ with Neumann boundary condition. (English) Zbl 0735.35063
The authors consider the problem; $-\Delta u=u^{(n+2)/(n- 2)}+\lambda\alpha u \text{ in } \Omega;\;u>0;\quad {\partial u \over \partial\nu}=0 \text{ on } \partial\Omega, \tag{*}$ where $$\alpha\in C^ \infty(\bar\Omega)$$ and $$\Omega$$ is a bounded domain of $$\mathbb{R}^ n$$, $$n\geq4$$ with regular boundary.
Under some assumptions on the function $$\alpha$$ and on the ”flatness” of the boundary $$\partial\Omega$$ the authors prove that problem (*) admits a solution $$u\in C^ 2(\bar\Omega)$$ if and only if $$\lambda$$ belongs to a suitable interval $$(\nu,\lambda(\alpha))$$.
Reviewer: M.A.Vivaldi (Roma)

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs