It is not exactly a probability question. This is part of my research where I need to find the intersection of the 2 curves for 0<p<1 given a fixed n and k. I am not majoring in mathematics and hence am having problems with solving it.
I don't think that there are multiple solutions as you have pointed out. In particular, (1) is monotonically increasing while (2) is a lateral reflection of it which is monotonically decreasing. So there must a a unique p that solves it.
By the way, I suspect that p=(k-1)/(n-1) from a series of graphs I have plotted but just can seem to prove it analytically.