  In this paper we consider the isentropic compressible Euler equations in two
space dimensions together with particular initial data. The latter consists
only of two constant states, where one state lies on the lower and the other
state on the upper half plane. The aim is to investigate if there exists a
unique entropy solution or if the convex integration method produces infinitely
many entropy solutions. For some initial states this question has been answered
by E. Feireisl and O. Kreml, and also G.-Q. Chen and J. Chen, where there
exists a unique entropy solution. For other initial states E. Chiodaroli, O.
Kreml and C. De Lellis showed that there are infinitely many entropy solutions.
For still other initial states the question on uniqueness remained open and
this will be the content of this paper. This paper can be seen as a completion
of the aforementioned papers by showing that the solution is non-unique in all
cases (except if the solution is smooth).
