Why is lexington being searched?

Hamlet's fourth soliloquy is probably his most famous one - "to be or not to be."it is rather different from his previous soliloquies because now he is contemplating something that didn't come up before - suicide. he is thinking whether it is worth it to fight to find his father's murderer when in the end all of them are going to die, so why not die sooner than later. he is contemplating if he should go on fighting or just kill himself and end it all, which is why this soliloquy in particular is rather depressing, hesitant, and gives a deeper insight into hamlet's madness.