Some popular party balloons, used to make balloon animals, have the property that the balloon inflates along part of its length, while the rest remains uninflated. In spite of the extensive literature on phase transitions in solid, this particular problem has not received much attention in the literature. In this paper, an infinite cylindrical membrane is analyzed. The existence of heteroclinic solutions which connect a uniformly inflated state at minus infinity to a different uniformly inflated state at plus infinity is established.