Nonlinear elliptic equations and intrinsic potentials of Wolff type

Abstract

We give necessary and sufficient conditions for the existence of weak solutions to the model equation $$-{\Delta }_{p}u=\sigma u^q\text{on}{\mathbb{R}}^n,$$ in the case $0<q<p-1$, where $\sigma \ge 0$ is an arbitrary locally integrable function, or measure, and ${\Delta }_{p}u=\mathrm{div}(\nabla u|\nabla u{|}^{p-2})$ is the $p$-Laplacian. Sharp global pointwise estimates and regularity properties of solutions are obtained as well. As a consequence, we characterize the solvability of the equation $$-{\Delta }_{p}v=b\frac{|\nabla v{|}^p}{v}+\sigma \text{on}{\mathbb{R}}^n,$$ where $b>0$. These results are new even in the classical case $p=2$. Our approach is based on the use of special nonlinear potentials of Wolff type adapted for "sublinear" problems, and related integral inequalities. It allows us to treat simultaneously several problems of this type, such as equations with general quasilinear operators $\text{div}\mathcal{A}(x,\nabla u)$, fractional Laplacians $(-\Delta {)}^{\alpha }$, or fully nonlinear $k$-Hessian operators.

47 pages

References in corpus (3)

• Finite energy solutions of quasilinear elliptic equations with sub-natural growth terms
• Local and global behaviour of nonlinear equations with natural growth terms
• p-Laplacian type equations involving measures

Cited by in corpus (12)

• Pointwise estimates of solutions to semilinear elliptic equations and inequalities
• Pointwise estimates of solutions to nonlinear equations for nonlocal operators
• Pointwise estimates of Brezis-Kamin type for solutions of sublinear elliptic equations
• A sublinear version of Schur's lemma and elliptic PDE
• Finite energy solutions to inhomogeneous nonlinear elliptic equations with sub-natural growth terms
• 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
• Trace inequalities of the Sobolev type and nonlinear Dirichlet problems
• Minimal L^p-Solutions to Singular Sublinear Elliptic Problems
• 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