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Existence and nonexistence of interior-peaked solution for a nonlinear Neumann problem. (English) Zbl 1140.35440
Summary: We show that the critical problem \[ \begin{split} -\Delta u+\lambda u&=|u|^{2^*-2}u+a|u|^{q-2}u \quad\text{in }\Omega,\\ \frac{\partial u}{\partial\nu}&=0\quad\text{on }\partial\Omega, \end{split} \]
where \(\Omega\) is a smooth, bounded domain in \(\mathbb{R}^N\) (\(N\geq3\)), \(\lambda>0\) and \(a\geq 0\) are constants, \(2^*=2N/(N-2)\) and \(2<q< 2N/(N-2)\), has no positive solutions concentrating, as \(\lambda\to\infty\), at interior points of \(\Omega\) if \(a = 0\), but for a class of symmetric domains \(\Omega\), the problem with \(a > 0\) has solutions concentrating at interior points of \(\Omega\).

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
35J20 Variational methods for second-order elliptic equations
35J60 Nonlinear elliptic equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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