Hi @fadi12345
Your children are entitled to come to Germany as the children of a German citizen. They apply for a visa from the German embassy in Beirut and once in Germany apply for residence permits from your local Ausländerbehörde.
The situation with their mother is more complicated. She can apply for a residence permit as a parent of a child who has a residence permit in Germany. However, this is only allowed in cases where it is necessary to avoid what is called extraordinary hardship. She will have to show that her situation is such a case when making her application at the embassy. It is not enough to mention the separation from her children or the general situation in Lebanon. Her personal circumstances must be such that to deny her a visa would cause this extraordinary hardship.
In addition, the mother has to prove she can be financially self-sufficient in Germany.
Best,
Éanna