Self-focusing Multibump Standing Waves in Expanding Waveguides

Let M be a smooth k-dimensional closed submanifold of $${\mathbb{R}^N, N \geq 2}$$, and let ΩR be the open tubular neighborhood of radius 1 of the expanded manifold $${M_R := \{R_x : x \in M\}}$$. For R sufficiently large we show the existence of positive multibump solutions to the problem$$ -\Delta u + \lambda u = f(u)\,{\rm in}\,\Omega_R,\quad u= 0\,{\rm on}\,\partial\Omega_R. $$The function f is superlinear and subcritical, and λ >  −λ1, where λ1 is the first Dirichlet eigenvalue of −Δ in the unit ball in $${\mathbb{R}^{N-k}}$$.