Alternating sign multibump solutions of nonlinear elliptic equations in expanding tubular domains

Abstract

Let $\Gamma$ denote a smooth simple curve in ${\mathbb{R}}^N$, $N\ge 2$, possibly with boundary. Let ${\Omega }_R$ be the open normal tubular neighborhood of radius 1 of the expanded curve $R\Gamma :=\{Rx\mid x\in \Gamma \setminus \partial \Gamma \}$. Consider the superlinear problem $-\Delta u+\lambda u=f(u)$ on the domains ${\Omega }_R$, as $R\to \infty$, with homogeneous Dirichlet boundary condition. We prove the existence of multibump solutions with bumps lined up along $R\Gamma$ with alternating signs. The function $f$ is superlinear at 0 and at $\infty$, 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.