My question is to Maximise $(a+b)^p+(a+c)^p+(b+c)^p$ subject to $a^{p'}+b^{p'}+c^{p'}=1$. Here, $a,b,c>0$, $p\in (1,\infty)$ and $p'$ is the dual exponent of $p$.
I tried Lagrange's multiplier method, but I cannot solve the complicated equation if $p\neq 2$.
Moreover, it is unknown to me what range of $p$ will ensure that a maximum can be attained. If possible, I would like to figure out the range.