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A class of nonlinear conservative elliptic equations in cylinders. (English) Zbl 0918.35051
The authors investigate the asymptotic behaviour of solutions of \[ u_{tt} + \Delta_g u - \lambda u + | u| ^{q-1}u = 0 \quad\text{in } M\times [0,\infty) \tag{\(*\)} \] where \((M,g)\) is a compact Riemannian manifold without boundary, \(\Delta_g\) is the Laplace-Beltrami operator on \(M\) and \(q>1\) and \(\lambda\) are constants. This study is motivated in part by questions about the asymptotic behaviour of solutions of the conformally invariant Emden-Fowler equation \[ -\Delta u + c| x| ^{-2}u = u^{(N+2)/(N-2)} \quad\text{in} {\mathbb{R}}^N - \{0\}, \] which after a suitable change of variables can be put into the form \((*)\) with \((M,g)\) equal to the standard sphere \({\mathbb{S}}^{N-1}\). The authors formulate a number of conditions under which solutions of the stationary problem \[ \Delta_g u - \lambda u + | u| ^{q-1}u = 0 \quad\text{in} M \] are constant, and then examine under what conditions solutions of \((*)\) asymptotically approach solutions of the homogeneous problem \[ \phi_{tt} - \lambda\phi + | \phi| ^{q-1}\phi = 0. \] The existence of solutions of \((*)\) with given initial condition \(u(\cdot,0)=u_0\) and vanishing at infinity is also proved. In the final section some partially homogenized variants of \((*)\) are considered in the case \(q=3\).

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
35B40 Asymptotic behavior of solutions to PDEs
58J32 Boundary value problems on manifolds
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