A Liouville type theorem for some critical semilinear elliptic equations on noncompact manifolds

Abstract

Consider the equation Δu-Vu+Wu ( n + 2 ) / ( n - 2 ) = 0 on certain noncompact manifolds with nonnegative Ricci curvature. Assuming V = V(x) > 0 decays slower than inverse square of the distance and W is bounded between two positive constants, we prove that any finite energy nonnegative solutions to the above equations must have exponential decay near infinity. Generalizing some well-known obstruction (Kazdan-Warner and Bourguignon-Ezin) from compact manifolds to the noncompact ones, we also show that these solutions must be zero under additional conditions. The above condition on V is essentially sharp.