Question

why did Lincoln declared a war in the southern states?

Answer 1

Under the War Powers clause of the Constitution, only Congress has the power to declare war. Source: War Powers Clause

Your question, at this point, is  non sequitur.  Lincoln never declared war on the South, because he had no Constitutional power to do so.

Under the Constitution, the President has the authority to put down armed insurrection. The War Powers Act of 1807 requires him to use the state militia as a first resort, though. Source: Insurrection Act

There is no question that what happened at Fort Sumter amounted to armed insurrection.

Another thing: war can be declared only between bodies that are recognized under international law as sovereign states. Source: Declaration of war

The Confederate States of America were never recognized as a sovereign state not only by the United States but also by any sovereign state on planet Earth. It is non sequitur that the United States can declare war on any entity that is not recognized under international law as a sovereign state.

I have to say that it's not an everyday event on quora that a single question question includes two non sequiturs.