We investigate the energy nondecay and existence of scattering states for solutions to the initial-boundary-value problem for the nonlinear wave equation in exterior domains. When the space dimension is odd, the domain meets no geometrical condition. Otherwise, we assume that the obstacle is convex. For odd-dimensional general domains, taking into account the effective dissipation in trapping regions, we can derive the existence of scattering states. In particular, we can obtain also an $L^2$ bound of solutions. The method in deriving the energy nondecay is to utilize Huyghens' principle. For even-dimensional domains outside the convex obstacle, the asymptotics stated in the odd-dimensional case are also valid.

We prove results on the existence of extremal functions for critical Sobolev inequalities on Riemannian manifolds when the functions are invariant under an isometry group. In order to get those results, we study precisely a concentration phenomenon around an orbit for a sequence of solutions of a nonlinear PDE invariant under the isometry group.

The uniqueness and profile of solutions for a Matukuma-type equation on the whole space as the power of the nonlinear term tends to one are discussed. By using information on a linear eigenvalue problem and Lyapunov-Schmidt-like reduction, we obtain the uniqueness and asymptotic profile of solutions. We also consider the case of bounded domain in the same manner.

This paper is concerned with bifurcation problems for nonlinear partial differential equations of the form $$-\mbox{div}(a(|\nabla u|)\nabla u) = \lambda g(u)$$ which are subject to Dirichlet boundary conditions. We show the existence of infinitely many nontrivial solutions of the eigenvalue problems in the case where $a(|t|) = |t|^{p-2}$ and $g(t) = |t|^{p-2}t, $ $p> 1.$ More general situations are also considered.