When considering Lagrange's method of multipliers for finding maximal solutions to a set of non-linear equations, I have reached a set of 4 equations in 4 real unknowns, $(a,b,c,\lambda)$:
$(a+b)^2+(a+c)^2 = -(8/3)\lambda a$
$(b+c)^2+(b+a)^2 = -(8/3)\lambda b$
$(c+a)^2+(c+b)^2 = -(8/3)\lambda c$
$a^2+b^2+c^2 = 1.$
$a,b,c>0$
Clearly $a=b=c=1/\sqrt 3$ and $\lambda = -\sqrt 3$ is a solution. Is there anything that can be said about the uniqueness of this without messy computational algebra. Is this due to the symmetric nature of the equations or for any other reasons?
You dont really have $4$ equations with $4$ unknowns. Simply subtract the third equation from the first equation and compare with the second. You will get $a=b+c$. To see if there is a unique solution or not, you need to have a single equation with a single parameter. Then if this function is monotone, then you are done. If not you will be sure that there exists as least one more solution. If equations are not reducable to a single equation, then it is difficult to show the uniqueness.