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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 \(\Omega\).

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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