Self-focusing Multibump Standing Waves in Expanding Waveguides

摘要

Let M be a smooth k-dimensional closed submanifold of R(N), N >= 2, and let Omega(R) be the open tubular neighborhood of radius 1 of the expanded manifold M(R) := {Rx : x is an element of M}. For R sufficiently large we show the existence of positive multibump solutions to the problem -Delta u + lambda u = f(u) in Omega(R), u = 0 on partial derivative Omega(R). The function f is superlinear and subcritical, and lambda > -lambda(1), where lambda(1) is the first Dirichlet eigenvalue of -Delta in the unit ball in R(N-k).