I fell down in my apartment comlez stairway because of no lights in the stairways. Are the responsible

They may be.  If it is their responsibility to maintain the hallways and if they had notice that the light was burned out prior to the time you fell, you may have a claim against them.  A personal injury is called a "tort" and you should look in your local area for an attorney who specializes in personal injury/tort cases.   Michael J. Katz