Alternating Sign Multibump Solutions of Nonlinear Elliptic Equations in Expanding Tubular Domains

Let Γ denote a smooth simple curve in ℝN, N ≥ 2, possibly with boundary. Let ΩR be the open normal tubular neighborhood of radius 1 of the expanded curve RΓ: = {Rx {pipe} x ∈ Γ{set minus}∂Γ}. Consider the superlinear problem - Δu + λu = f(u) on the domains ΩR, as R → ∞, with homogeneous Dirichlet boundary condition. We prove the existence of multibump solutions with bumps lined up along RΓ with alternating signs. The function f is superlinear at 0 and at ∞, but it is not assumed to be odd. If the boundary of the curve is nonempty our results give examples of contractible domains in which the problem has multiple sign changing solutions. © 2013 Copyright Taylor and Francis Group, LLC.