What is the distribution of $\Lambda$? 
As in Likelihood ratio test for $H_0:(\mu_1,\mu_2)=(0,0)$ vs $H_1:(\mu_1,\mu_2) \neq (0,0)$,
There are $X_1, X_2$ where $X_i \sim N(\mu_i,1), i=1,2$. They are independent. The question is


Find the likelihood ratio test with $H_0:(\mu_1,\mu_2)=(0,0), H_1:(\mu_1,\mu_2) \neq (0,0)$. The significance level is $\alpha (0< \alpha <1)$ and parameter space $\Omega$ is $$\Omega = \left\{ (\mu_1,\mu_2) : \mu_1 \geq 0, \mu_2 \geq 0\right\}$$


What is the distribution of $\Lambda$?

We have known that $$\Lambda = x_1^2I_{(x_1>0, x_2<0)} + x_2^2I_{(x_1<0, x_2>0)} + (x_1^2+x_2^2)I_{(x_1>0, x_2>0)}$$

You also know the distribution of $x_1$ and $x_2$ when $H_0$ is true. It's a bivariate normal distribution centered at $0,0$ and with variance $I$. Use that to get the distribution of $\Lambda$.
Because of the indicator functions you will get to handle it as a mixture distribution. In 25% of the cases you get $I_{(x_1>0, x_2<0)} = 1$, in 25% of the cases you get $I_{(x_1<0, x_2>0)} = 1$, in 25% of the cases you get $I_{(x_1>0, x_2>0)} = 1$, and in 25% of the cases $\Lambda = 0$.
Below is an illustration that may help to visualize it. There's $10^4$ points simulated that illustrate the bivariate distribution of $x_1,x_2$. Two example points are drawn with isolines for which the value of $\Lambda$ is constant.

So for example the probability that $P(\Lambda \leq 1.92)$ consist of the sum of $$P(\Lambda \leq 1.92) =  
\underbrace{P(x_1<=0,x_2<=0)}_{= 0.25}+
\underbrace{P(x_1>0,x_2<=0 \text{ and } x_1^2 \leq 1.92)}_{= 0.25 \times P(\chi_1^2 <1.92)}+
\underbrace{P(x_1<=0,x_2>0  \text{ and } x_2^2 \leq 1.92 )}_{= 0.25 \times P(\chi_1^2 <1.92)}+
\underbrace{P(x_1>0,x_2>0  \text{ and } x_1^2 + x_2^2 \leq 1.92)}_{= 0.25 \times P(\chi_2^2 <1.92)}  $$
