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In general the answer is "no." Take for example $\Omega = (-\pi/2,\,\pi/2) \subset \mathbb{R}$, 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 $a_1 \neq a_2$.