Computation of the difference of topology at infinity for Yamabe-type problems on annuli-domains. I, II
For \(\varepsilon> 0\) let \(A_\varepsilon= \{x\in\mathbb{R}^n; \varepsilon<|x|< 1/\varepsilon\}\), \(n\geq 3\). The authors consider the nonlinear elliptic problem \[ -\Delta u= u^{(n+2)/n-2},\quad u>0\quad\text{on }A_\varepsilon,\quad u= 0\quad\text{on }\partial A_\varepsilon.\tag{P\(_\varepsilon\)} \] The positive critical points of the functional \(J_\varepsilon\) on \(H^1_0(A_\varepsilon)\), \[ J_\varepsilon(u)= (1/2) \int_{A_\varepsilon}|\nabla u|^2- {n-2\over 2n} \int_{A_\varepsilon}|u|^{2n/n- 2}, \] are solutions of \((\text{P}_\varepsilon)\). This problem is delicate from a variational viewpoint because of the possible existence of critical points at infinity, which are orbits of \(J_\varepsilon\) along which \(J_\varepsilon\) remains bounded, the gradient goes to zero, and the orbits do not converge. To find the solutions of \((\text{P}_\varepsilon)\) by studying the difference of topology between the level sets of \(J_\varepsilon\), it becomes essential to evaluate the topological contribution of the critical points at infinity. In the first part of this work one computes the difference of topology at infinity in the particular case of double blow-up for thin annuli-domains. In the second part one computes it for expanding annuli \((\varepsilon\to 0)\).
scientific article; zbMATH DE number 1425141