The problem I am considering is the following:
\begin{align*} &\max_{p = (p_1, p_2,\cdots, p_N)} \int_{v} \sum_{i=1}^{N} v_i p_i(v) dv - \int_{v} \sum_{i=1}^{N} \frac{ |\nabla p_i(v)|^2}{p_i(v)}dv \\ & s.t. \quad p_i(v) >0 \;\; \forall i \;\; \forall v \in [0,1]^n\\ & \quad \quad \sum_{i=1}^N p_i(v) =1 \;\; \forall i \;\; \forall v \in [0,1]^n \end{align*} where $v=(v_1,v_2,\cdots,v_N) \in [0,1]^N$, and $p(v)=(p_1(v),p_2(v),\cdots,p_N(v))$ is a mapping from $[0,1]^N$ to $[0,1]^N$, with the constraint that $\sum_{i=1}^N p_i(v) =1$, we can also think that $p(v)$ is a discrete distribution of support size $N$ . And $\nabla p_i(v)=(\frac{\partial p_i(v)}{\partial v_1},\frac{\partial p_i(v)}{\partial v_2},\cdots,\frac{\partial p_i(v)}{\partial v_N})$. The term $\sum_{i=1}^{N} \frac{ |\nabla p_i(v)|^2}{p_i(v)}$ is the fisher information of the family of the discrete distribution $\{p(v)\}_{v}$, and throughout we assume $p$ satisfies the regularity condition to ensure the fisher information is well-defined.
I am looking for help on whether my derivation of the Euler-Lagrange equation is correct and help on solving the corresponding equation if it is correct.
I am assuming that each $p_i$ is a $\mathcal{C}^2$ function, and consider the Lagrangian $J(p)=\int_{v} \sum_{i=1}^{N} [v_i p_i(v) - \frac{ |\nabla p_i(v)|^2}{p_i(v)}] dv+ \int_{v} \mu(v)(1-\sum_{i=1}p_i(v)) dv$, and the function $g(\lambda) = J(p^* + \lambda h) $ where $h$ is a commonly supported smooth function that vanish on the boundary of $[0,1]^N$. Since the function $g(\lambda)$ is maximized at $0$, we hence have the necessary condition $g'(0)=0$, which implies that \begin{align*} 0 &= \int_v \sum_i^N [v_i h_i(v)+ \frac{ |\nabla p^*_i(v)|^2}{p^*_i(v)^2}h_i(v) - \frac{\langle \nabla p_i^*(v) , \nabla h_i(v) \rangle }{p^*_i(v)} - \mu(v) h_i(v) ]dv\\ &=\int_v \sum_i^N [v_i h_i(v)+ \frac{ |\nabla p^*_i(v)|^2}{p^*_i(v)^2}h_i(v) - \mu(v) h_i(v) ]dv -\int_v \textbf{div} (h_i(v) \frac{\nabla p^*_i(x)}{p_i^*(v)})- h_i(x) \textbf{div} (\frac{\nabla p^*_i(v) }{p_i^*(v)}) dv & (\text{Integration by parts})\\ &=\int_v \sum_i^N [v_i h_i(v)+ \frac{ |\nabla p^*_i(v)|^2}{p^*_i(v)^2}h_i(v) - \mu(v) h_i(v) ]dv - \int_{\partial [0,1]^n} h_i(x) \frac{\nabla p_i^*(v)}{p_i^*(v)} \nu dS - \int_v h_i(x) \textbf{div} (\frac{\nabla p^*_i(v)}{p_i^*(v)}) dv & (\text{Divergence theorem})\\ &= \int_v \sum_i^N [v_i h_i(v)+ \frac{ |\nabla p^*_i(v)|^2}{p^*_i(v)}h_i(v) - \mu(v) h_i(v) +h_i(x) \textbf{div} (\frac{\nabla p^*_i(v)}{p_i^*(v)}) dv & (h_i \text{ vanishes at the boundary})\\ &=\int_v \sum_i^N h_i(v)[v_i + \frac{ |\nabla p^*_i(v)|^2}{p^*_i(v)^2} - \mu(v) +( \frac{\sum_{j}^{n}(\frac{\partial p^*_i(v)}{\partial v_j})^2}{p_i^*(v)^2} -\frac{\sum_{j}^{n}\frac{\partial^2 p^*_i(v)}{\partial v_j^2}}{p_i^*(v)} )] dv\\ &=\int_v \sum_i^N h_i(v)[v_i + 2\frac{ |\nabla p^*_i(v)|^2}{p^*_i(v)^2} - \mu(v) -\frac{\Delta p_i^*(v)}{p_i^*(v)} ] dv \end{align*}
Since any the derivation holds for any smooth and compactly supported perturbation $h$, we have: $$v_i + 2\frac{ |\nabla p^*_i(v)|^2}{p^*_i(v)^2} - \mu(v) -\frac{\Delta p_i^*(v)}{p_i^*(v)} =0 \quad \quad \forall i, \forall v$$
