JournalsrlmVol. 30, No. 4pp. 733–758

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

  • Igor E. Verbitsky

    University of Missouri, Columbia, USA
Quasilinear elliptic equations with sub-natural growth terms and nonlinear potential theory cover
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Abstract

We discuss recent advances in the theory of quasilinear equations of the type Δpu=σuq-\Delta_{p} u = \sigma u^{q} in Rn\mathbb R^n, in the case 0<q<p10 < q < p-1, where σ\sigma is a nonnegative measurable function, or measure, for the pp-Laplacian Δpu=div(up2u)\Delta_{p}u= \mathrm{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 uBMO(Rn),uLlocr(Rn)u \in \mathrm{BMO}(\mathbb R^n), u \in L^r_{{\mathrm loc}}(\mathbb R^n), etc., and prove an enhanced version of Wolff's inequality for intrinsic nonlinear potentials associated with such problems.

Cite this article

Igor E. Verbitsky, Quasilinear elliptic equations with sub-natural growth terms and nonlinear potential theory. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 30 (2019), no. 4, pp. 733–758

DOI 10.4171/RLM/869