Solutions of semilinear elliptic equations in tubes

Abstract

Given a smooth compact k-dimensional manifold Λembedded in $\mathbb {R}^m$, with m\geq 2 and 1\leq k\leq m-1, and given ε>0, we define B_ε(Λ) to be the geodesic tubular neighborhood of radius εabout Λ. In this paper, we construct positive solutions of the semilinear elliptic equation Δu + u^p = 0 in B_ε(Λ) with u = 0 on \partial B_ε(Λ), when the parameter εis chosen small enough. In this equation, the exponent p satisfies either p > 1 when n:=m-k \leq 2 or p\in (1, \frac{n+2}{n-2}) when n>2. In particular p can be critical or supercritical in dimension m\geq 3. As εtends to zero, the solutions we construct have Morse index tending to infinity. Moreover, using a Pohozaev type argument, we prove that our result is sharp in the sense that there are no positive solutions for p>\frac{n+2}{n-2}, n\geq 3, if εis sufficiently small.