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We study quasilinear elliptic equations of the type $$-\Delta_pu=\sigma \,u^q \quad \text{in} \, \, \, \mathbb{R}^n,$$ where $\Delta_p u=\nabla\cdot(\nabla u |\nabla u|^{p-2})$ is the $p$-Laplacian (or a more general$\mathcal{A}$-Laplace operator $\text{div} \, \mathcal{A}(x, \nabla u)$), $0<q< p-1$, and $\sigma \ge 0$ is an arbitrary locally integrable function ormeasure on $\mathbb{R}^n$.  We obtain necessary and sufficient conditions for the existence of positivesolutions (not necessarily bounded) which satisfy global pointwise estimates ofBrezis-Kamin type given in terms of Wolff potentials. Similar problems with thefractional Laplacian $(-\Delta )^{\alpha}$ for $0<\alpha<\frac{n}{2}$ aretreated as well, including explicit estimates for radially symmetric $\sigma$.Our results are new even in the classical case $p=2$ and $\alpha=1$.

Pointwise estimates of Brezis-Kamin type for solutions of sublinear elliptic equations