The solutions to the Schrödinger equation are not unique in general, and uniqueness depends on several things such as the form of the potential and boundary conditions. Many papers have discussed uniqueness of solutions to the Schrödinger equation for specific classes of potentials and boundary conditions, but in general it is possible to come up with cases where the solution is not unique.
In fact for most potentials and boundary conditions, the Schrödinger equation has several solutions (a ground state solution, and several excited state solutions).
Is at least the ground state solution unique?
For many "physically realistic" potentials and boundary conditions that we normally use (for example, Morse potentials and generalizations of it) we do have a unique ground state energy, but keep in mind that multiple wavefunction solutions can have the same ground state energy: These are called degenerate solutions.
What about DFT and the Hohenberg-Kohn Theorem of Existence?
The theorem states that "the external potential (and hence the total energy), is a unique functional of the electron density." This means that for a given density $\rho$, there is a unique energy functional $E[\rho(r)]$.
Let's say there's a degenerate energy level that has two degenerate solutions to the Schrödinger equation, with densities $\rho_1(r)$ and $\rho_2(r)$. There is a unique energy functional of $\rho_1(r)$, which is $E_1[\rho_1(r)]$, and a unique functional of $\rho_2(r)$, which is $E_2[\rho_2(r)]$. When these two functionals are applied, the resulting energy is the same either way (because the energy is degenerate), but that doesn't mean the functionals are the same or the densities are the same.
An energy can correspond to two different solutions of the Schrödinger equation, each with their own unique functional that gives that energy. Since these two solutions of the Schrödinger equation are not the same, the solutions are not unique.
