The following problem appears in a model that I developed to try to find the optimal mechanism by which an international institution could reward developing countries for their efforts to subsidize renewable energies. We have an exogenously given probability density function $f(x,y)$ in whose support we have that $x>0$ and $y>1$. The problem consists of finding the real-valued functions $s(x,y), R(x,y)$ that maximize: \begin{equation} \int_{x=0}^{\infty} \int_{y=1}^{\infty} s(x,y)^{\frac{1}{y-1}} x f(x,y) \quad dy dx \end{equation} subject to the following three constraints. Firstly, there is a PDE constraint which must hold $\forall x>0$ and $\forall y>1$: \begin{equation} \frac{-x}{(y-1)} s^{\frac{1}{y-1}} (\frac{1}{(y-1)^{2}}log(s) \frac{\partial s}{\partial x} +\frac{\partial s}{\partial y})+ \frac{1}{(y-1)^{2}}log(s) \frac{\partial R}{\partial x}+\frac{\partial R}{\partial y}=0 \end{equation} Secondly, there is the following pointwise constraint, which must also hold $\forall x>0$ and $\forall y>1$ \begin{equation} R(x,y) \geq \frac{x}{y} s(x,y)^{\frac{y}{y-1}} \end{equation} Thirdly, there is the following constraint: \begin{equation} \int_{x=0}^{\infty} \int_{y=1}^{\infty} R(x,y) f(x,y) \quad dy dx =G \end{equation} where $G$ is an exogenously given positive number.
Is there any literature studying this kind of problem? Is there any way to reduce the problem to a set of differential equations like the Euler-Lagrange equations in the calculus of variations or the Hamilton-Jacobi-Bellman equations in control theory?