Solutions of Semilinear Elliptic Equations in Tubes

Given a smooth compact k-dimensional manifold Λ embedded in ? ^m , with m≥2 and 1≤k≤m?1, and given ?>0, we define B _? (Λ) to be the geodesic tubular neighborhood of radius ? about Λ. In this paper, we construct positive solutions of the semilinear elliptic equation $$begin{cases} Delta u + u^p = 0 &mbox{in } B_{epsilon}(varLambda) u = 0 & mbox{on } partial B_{epsilon}(varLambda) , end{cases} $$ when the parameter ? is chosen small enough. In this equation, the exponent p satisfies either p>1 when n:=m?k≤2 or $pin(1, frac{n+2}{n-2})$ when n>2. In particular, p can be critical or supercritical in dimension m≥3. As ? 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 $p>frac{n+2}{n-2}$ , n≥3, if ? is sufficiently small.