Cao, Daomin; Noussair, E. S.; Yan, Shusen Existence and nonexistence of interior-peaked solution for a nonlinear Neumann problem. (English) Zbl 1140.35440 Pac. J. Math. 200, No. 1, 19-41 (2001). 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 \(\Omega\). Cited in 15 Documents 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 PDFBibTeX XMLCite \textit{D. Cao} et al., Pac. J. Math. 200, No. 1, 19--41 (2001; Zbl 1140.35440) Full Text: DOI