In general the answer is "no." Let $\Omega = (-\pi/2,\,\pi/2) \subset \mathbb{R}$, and for $\varphi \in C^{\infty}_0(\Omega)$ take
$$u_1 = \cos(x), \quad u_2 = \cos(x) + \epsilon \varphi$$
$$a_1 = 1, \quad a_2 = \frac{\cos(x) + \epsilon \varphi}{\cos(x) - \epsilon \varphi''},$$
$$\lambda = 1.$$
For $\epsilon > 0$ small, the desired conditions are satisfied, but $a_1 \neq a_2$.