On some nonlinear boundary-value problems in cylindrical domains (Q1280074)

The author considers a linear elliptic second-order equation in divergent form in a semi-infinite \(n\)-dimensional cylinder \(\Pi_0=\Omega\times [0,\infty)\), where \(\Omega\subset \mathbb{R}^{n-1}\) is a Lipschitzian bounded domain, and \(0\leq x_n <\infty\). The solution satisfies the nonlinear boundary condition \(\partial u/\partial v+a_0| u|^{q-1}u=0\), where \(\partial/ \partial v\) denotes the differentiation along the external conormal, \(a_0=\text{const}>0\), \(q=\text{const}\); it is also assumed that \(\lim_{x_n\to \infty} u(x)=0\). The author proves that \(\lim_{x_n\to\infty} x_n^{2/(q-1)} u(x)\) exists and equals one of the three numbers \(0,c,-c\) \((c= \text{const})\); moreover, if \(\lim_{x_n\to \infty} x_n^{2/(q-1)} u(x)=0\), then the solution \(u(x)\) decays exponentially. The author proves also the nonexistence of positive solutions for \(a_0<0\), \(q>1\), and investigates the asymptotic behaviour of oscillating solutions.