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 Ïμ>0, we define B _Ïμ (Λ) to be the geodesic tubular neighborhood of radius Ïμ about Λ. In this paper, we construct positive solutions of the semilinear elliptic equation {δu + u^/ = 0 in B_ε (λ) u = 0 on ∂ B_ε (λ), when the parameter Ïμ is chosen small enough. In this equation, the exponent p satisfies either p>1 when n:=m-k≤2 or p ε (1, n+2/n-2) when n>2. In particular, p can be critical or supercritical in dimension m≥3. As Ïμ 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 p > n+2/n-2, n≥3, if Ïμ is sufficiently small.