Interaction between the Geometry of the Boundary and Positive Solutions of a Semilinear Neumann Problem with Critical Nonlinearity

✍ Scribed by F. AdimurthiPacella; S.L. Yadava

• Publisher: Elsevier Science
• Year: 1993
• Tongue: English
• Weight: 837 KB
• Volume: 113
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1993.1053

✦ Synopsis

We consider the problem: (-\Delta u+\lambda u=u^{(n+2) /(n-2)}, u>0) in (\Omega, \partial u / \hat{\partial} v=0) on (\partial \Omega), where (\Omega) is a bounded smooth domain in (\mathbb{R}^{n}(n \geqslant 3)). We show that, for (\lambda) large, least-energy solutions of the above problem have a unique maximum point (P_{i}) on (\partial \Omega) and the limit points of (P_{\lambda}), as (i \rightarrow \infty) are contained in the set of the points of maximum mean curvature. We also prove that, if (\partial \Omega) has (k) peaks then the equation has at least (k) solutions for (\lambda) large. (a^{2} 1993) Academic Press, Inc.