Solutions of semilinear elliptic equations in tubes

TL;DR

This paper constructs positive solutions for semilinear elliptic equations in tubular neighborhoods of manifolds, analyzing their existence, behavior as the neighborhood shrinks, and establishing sharp nonexistence results for supercritical exponents.

Contribution

It provides a method to construct solutions in small tubular neighborhoods for a range of nonlinearities, including critical and supercritical cases, and proves sharp nonexistence results.

Findings

Solutions exist for small ps when p is subcritical or critical.

Morse index of solutions tends to infinity as ps approaches zero.

No positive solutions for supercritical p when ps is sufficiently small.

Abstract

Given a smooth compact k-dimensional manifold \Lambda embedded in $\mathbb {R}^m$, with m\geq 2 and 1\leq k\leq m-1, and given \epsilon>0, we define B_\epsilon (\Lambda) to be the geodesic tubular neighborhood of radius \epsilon about \Lambda. In this paper, we construct positive solutions of the semilinear elliptic equation \Delta u + u^p = 0 in B_\epsilon (\Lambda) with u = 0 on \partial B_\epsilon (\Lambda), when the parameter \epsilon is 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 \epsilon tends 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 \epsilon is sufficiently small.