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

Abstract

We study quasilinear elliptic equations of the type $$-{\Delta }_{p}u=\sigma u^q\text{in}{\mathbb{R}}^n,$$ 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 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 $(-\Delta {)}^{\alpha }$ 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 $\alpha =1$.

24 pages

References in corpus (2)

• Finite energy solutions of quasilinear elliptic equations with sub-natural growth terms
• Two weight norm inequalities for vector-valued operators

Cited by in corpus (8)

• Pointwise estimates of solutions to nonlinear equations for nonlocal operators
• Pointwise estimates of solutions to semilinear elliptic equations and inequalities
• A sublinear version of Schur's lemma and elliptic PDE
• Quasilinear elliptic equations with sub-natural growth terms in bounded domains
• Existence of minimal solutions to quasilinear elliptic equations with several sub-natural growth terms
• The Dirichlet problem for sublinear elliptic equations with source
• Nonlocal Sublinear Elliptic Problems Involving Measures
• Finite energy solutions of quasilinear elliptic equations with Orlicz growth