Quasilinear elliptic equations with sub-natural growth terms and nonlinear potential theory

Abstract

We discuss recent advances in the theory of quasilinear equations of the type $-{\Delta }_{p}u=\sigma u^q\text{in}{\mathbb{R}}^n,$ in the case $0<q<p-1$, where $\sigma$ is a nonnegative measurable function, or measure, for the $p$-Laplacian ${\Delta }_{p}u=\text{div}(|\nabla u{|}^{p-2}\nabla u)$, as well as more general quasilinear, fractional Laplacian, and Hessian operators. Within this context, we obtain some new results, in particular, necessary and sufficient conditions for the existence of solutions $u\in \text{BMO}({\mathbb{R}}^n)$, $u\in L_{\mathrm{loc}}^r({\mathbb{R}}^n)$, etc., and prove an enhanced version of Wolff's inequality for intrinsic nonlinear potentials associated with such problems.

Lemma 3.1 and related results extended, references [Mi1], [Mi2] added

Cited by in corpus (1)

• Existence of minimal solutions to quasilinear elliptic equations with several sub-natural growth terms