Under what conditions will a solution $y \in \Re^n$ exist for this system of nonlinear equations? If it exists, will it be unique?
$$ \mu_k = \int_0^\infty g_k(x) \, f(x, y) \, dx \qquad \forall \, k = 1, \dotsm, n $$
where $\mu_k$ are constants and all functions here are continuous.
If we cannot conclude anything in general, then consider a specific case in probability theory:
$$ \begin{align} \mu_k &= \frac{1}{Z(y)} \int_0^\infty g_k(x) \, e^{f(x, y)} \, dx \qquad \forall \, k = 1, \dotsm, n \\[1ex] Z(y) &= \int_0^\infty e^{f(x, y)} \, dx \end{align} $$
where $Z(y)$ is a normalizing constant while $g_k$ and $f$ are piecewise linear in $x$. Their exact forms depend on some data, but the following is true regardless of the data:
$$ \lim_{x \rightarrow \infty} f(x, y) = \begin{cases} \infty & \text{if } \; c^T y > 0 \\ -\infty & \text{if } \; c^T y < 0 \end{cases} $$
where $c \in \Re^n$ is constant.
