How should we interpret a transition rule $\epsilon,b\to c$ in a Pushdown Automata as Sipser defined in his book?
Suppose that your PDA is reading $011$.

At the beginning, the PDA is at state $q_1$, and the stack is empty. The only move the PDA can make is to put $\$$ on the stack and to transition to state $q_2$.

Since the next symbol to be read is $0$, the only move the PDA can make now is the self-loop on $q_2$ which puts $0$ on the stack. The input is now $11$, and the stack is $0\$$ (top of stack first).

Since the next symbol to be read is $1$, the only move the PDA can make is to transition to $q_3$, erasing $0$ from the top of the stack. The input is now $1$, and the stack is $\$$.

Since the next symbol to be read is $1$, the PDA must move to $q_4$, erasing $\$$ from the stack. The input is still $1$, and the stack is empty.

At this point, the PDA is stuck. It cannot make any move. However, it hasn't read all the input yet. We conclude that there is no accepting computation for this input, and so the PDA doesn't accept $011$.