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

We study quasilinear elliptic equations of the type $$-{\mathrm{\Delta }}_{p}u=\sigma u^q\text{in}{\mathbb{R}}^n,$$ where $\Delta_p u=\nabla \cdot(\nabla u |\nabla u|^{p-2})$isthe$p$-Laplacian(oramoregeneral$$\mathcal{A}$$-Laplaceoperator$$\text{div}$ , $\mathcal{A}$(x, \nabla u)$),$0<q < p-1$,and$\sigma \ge 0$ is an arbitrary locally integrable function or measure on ${\mathbb{R}}^n$. We obtain necessary and sufficient conditions for the existence of positive solutions (not necessarily bounded) which satisfy global pointwise estimates of Brezis-Kamin type given in terms of Wolff potentials. Similar problems with the fractional Laplacian $(-\mathrm{\Delta }{)}^{\alpha }$ for $0<\alpha <\frac{n}{2}$ are treated as well, including explicit estimates for radially symmetric $\sigma$. Our results are new even in the classical case $p=2$ and $\alpha =1$.