Maximize[{ (Sqrt[2] + Sqrt[2] Sqrt[1 + Sqrt[1 - x^2 - y^2]] )*
Log[(Sqrt[2] + Sqrt[2] Sqrt[1 + Sqrt[1 - x^2 - y^2]] )],
x > 0 && y > 0}, {x, y}, Reals]
This returns the same function as output. I have no idea why. Sorry if it is too basic. I am very new to mathematica.