ON A YAMABE-TYPE PROBLEM ON A THREE-DIMENSIONAL THIN ANNULUS

We consider the problem: (P-epsilon) : -Delta u(epsilon) = u(epsilon)(5), u(epsilon) > 0 in A(epsilon); u(epsilon) = 0 on partial derivative A(epsilon), where {A(epsilon) subset of R-3 : epsilon > 0} is a family of bounded annulus-shaped domains such that A(epsilon) becomes "thin" as epsilon -> 0. We show that, for any given constant C > 0, there exists epsilon(0) > 0 such that for any epsilon < epsilon(0), the problem (P-epsilon) has no solution u(epsilon), whose energy, integral(A epsilon)vertical bar del u(epsilon)vertical bar(2), is less than C. Such a result extends to dimension three a result previously known in higher dimensions. Although the strategy to prove this result is the same as in higher dimensions, we need a more careful and delicate blow up analysis of asymptotic profiles of solutions u(epsilon) when epsilon -> 0.