Uniqueness results through a priori estimates. II: Dirichlet problem
The author establishes a priori bounds and uniqueness for positive solutions of \[ -\Delta u-au= u^{{n+2}\over{n-2}}\quad\text{in }\Omega, \qquad u|_{\partial\Omega}\equiv 0,\tag{1} \] for bounded domains \(\Omega\) in \({\mathbb R}^n\) (\(n\geq 3\)) with smooth boundary and \(0<a<\lambda_1\) (\(\lambda_1\) the principal eigenvalue of \(-\Delta\) on \(\Omega\) subject to \(u|_{\partial\Omega}\equiv 0\)). He shows: if \(u_a\) is a positive solution of (1) for \(0<a<\lambda_1\), then \(\|u_a\|_\infty\to 0\) as \(a\to\lambda_1\) and there exists a \(\underline a\in(0,\lambda_1)\) such that \(u_a\) is the only positive solution of (1) for \(\underline a<a<\lambda_1\). The proof relies on reasoning by contradiction, and buzzwords such as energy independent bounds and blow-up points may be an incentive to read the paper which extends classical work of \textit{F. V. Atkinson} and \textit{L. A. Peletier} [Nonlinear Anal., Theory Methods Appl. 10, 755-776 (1986; Zbl 0662.34024)] in the case of \(\Omega\) being the unit ball.
scientific article; zbMATH DE number 1506525