Solutions of Semilinear Elliptic Equations in Tubes

Given a smooth compact k-dimensional manifold Λ embedded in ℝ m, with m≥2 and 1≤k≤m-1, and given ε{lunate}>0, we define B ε{lunate}(Λ) to be the geodesic tubular neighborhood of radius ε{lunate} about Λ. In this paper, we construct positive solutions of the semilinear elliptic equation {Mathematical expression} when the parameter ε{lunate} is chosen small enough. In this equation, the exponent p satisfies either p>1 when n:=m-k≤2 or {Mathematical expression} when n>2. In particular, p can be critical or supercritical in dimension m≥3. As ε{lunate} tends to 0, 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 {Mathematical expression}, n≥3, if ε{lunate} is sufficiently small. © 2012 Mathematica Josephina, Inc