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Mean curvature and least energy solutions for the critical Neumann problem with weight. (English) Zbl 1097.35046
The author investigates the existence of solutions of the semilinear Neumann problem $-\Delta u + \lambda u = Q(x)u^{2^*-1} \text{ in } \Omega, \qquad {{\partial u}\over{\partial \nu}} = 0 \text{ on }\partial\Omega, \qquad u > 0 \text{ in } \Omega \tag{(*).}$ Here $$\lambda>0$$ is a parameter, $$\Omega$$ is a bounded domain in $${\mathbb R}^N$$, $$N\geq 3$$, with smooth boundary $$\partial\Omega$$, $$\nu$$ is the outer unit normal to $$\partial\Omega$$, $$2^*=2N/(N-2)$$ is the critical Sobolev exponent, and $$Q$$ is assumed to be positive and Hölder continuous on $$\bar\Omega$$. In a previous paper the author and M. Willem [Calc. Var. Partial Differ. Equ. 15, No.4, 421-431 (2002; Zbl 1221.35116)] proved the existence of least energy solutions under certain assumptions. They also showed that new phenomena can occur for nonconstant $$Q$$ compared to the case $$Q\equiv 1$$. In particular, for $$Q\equiv 1$$ least energy solutions always exist for large enough $$\lambda$$, while this is not always true for nonconstant $$Q$$.
The main results here complement some of the earlier ones. One of these states that for certain combinations of nonconstant functions $$Q$$ and domains $$\Omega$$, there exists $$\bar\lambda>0$$ such that $$S_\lambda<{S\over{2^{2/N}Q_m^{(N-2)/N}}}$$ for all $$0<\lambda<\bar\lambda$$ and $$S_\lambda={S\over{2^{2/N}Q_m^{(N-2)/N}}}$$ for $$\lambda\geq \bar\lambda$$. Furthermore, $$S_\lambda$$ is achieved if and only if $$\lambda\in(0,\bar\lambda]$$. Here $$Q_m=\max_{\partial\Omega}Q$$ and the quantities $$S_\lambda$$ and $$S$$ are defined by $S_\lambda = \inf \left\{ \int_\Omega (|\nabla u|^2+\lambda u^2)\,dx: u\in H^1(\Omega),\quad \int_\Omega Q(x)|u|^{2^*}\,dx = 1 \right\},$ $S = \inf \left\{ \int_{{\mathbb R}^N} |\nabla u|^2\,dx: u\in D^{1,2}({\mathbb R}^N),\quad \int_{{\mathbb R}^N} |u|^{2^*}\,dx = 1 \right\}.$ A consequence of this is that a related optimal Sobolev inequality has extremals for such $$Q$$ and $$\Omega$$.

##### MSC:
 35J60 Nonlinear elliptic equations 35B33 Critical exponents in context of PDEs 35J20 Variational methods for second-order elliptic equations 35J25 Boundary value problems for second-order elliptic equations 47J30 Variational methods involving nonlinear operators
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