Finite energy solutions to inhomogeneous nonlinear elliptic equations with sub-natural growth terms

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 $\alpha = 1$.