  We study the existence and concentration of positive and nodal solutions to a
Schr\"odinger equation in the presence of a shrinking self-focusing core of
arbitrary shape. Via a suitable rescaling, the concentration gives rise to a
limiting profile that solves a nonautonomous elliptic semilinear equation with
a sharp sign change in the nonlinearity. We characterize the (radial or
foliated Schwarz) symmetries and the (polynomial) decay of the least-energy
positive and nodal limiting profiles.
