Abstract We study quasilinear elliptic equations of the type - Δ p u = σ u q + μ {-\Delta_{p}u=\sigma u^{q}+\mu} in ℝ n {\mathbb{R}^{n}} in the case 0 < q < p - 1 {0<q<p-1} , where μ and σ are nonnegative measurable functions, or locally finite measures, and Δ p u = div ( | ∇ u | p - 2 ∇ u ) {\Delta_{p}u=\operatorname{div}(\lvert\nabla u\rvert^{p-2}\nabla u)} is the p-Laplacian. Similar equations with more general local and nonlocal operators in place of Δ p {\Delta_{p}} are treated as well. We obtain existence criteria and global bilateral pointwise estimates for all positive solutions u: u ( x ) ≈ ( 𝐖 p σ ( x ) ) p - q p - q - 1 + 𝐊 p , q σ ( x ) + 𝐖 p μ ( x ) , x ∈ ℝ n , u(x)\approx({\mathbf{W}}_{p}\sigma(x))^{\frac{p-q}{p-q-1}}+{\mathbf{K}}_{p,q}% \sigma(x)+{\mathbf{W}}_{p}\mu(x),\quad x\in\mathbb{R}^{n}, where 𝐖 p {{\mathbf{W}}_{p}} and 𝐊 p , q {{\mathbf{K}}_{p,q}} are, respectively, the Wolff potential and the intrinsic Wolff potential, with the constants of equivalence depending only on p, q, and n. The contributions of μ and σ in these pointwise estimates are totally separated, which is a new phenomenon even when p = 2 {p=2} .