  We obtain necessary and sufficient conditions for the existence of a positive
finite energy solution to the inhomogeneous quasilinear elliptic equation \[
-\Delta_{p} u = \sigma u^{q} + \mu \quad \text{on} \;\; \mathbb{R}^n \] in the
sub-natural growth case $0<q<p-1$, where $\Delta_{p}$ ($1<p<\infty$) is the
$p$-Laplacian, and $\sigma$, $\mu$ are positive Borel measures on
$\mathbb{R}^n$. Uniqueness of such a solution is established as well.
  Similar inhomogeneous problems in the sublinear case $0<q<1$ are treated for
the fractional Laplace operator $(-\Delta)^{\alpha}$ in place of $-\Delta_{p}$,
on $\mathbb{R}^n$ for $0<\alpha<\frac{n}{2}$, and on an arbitrary domain
$\Omega \subset \mathbb{R}^n$ with positive Green's function in the classical
case $\alpha = 1$.
