Minimal L^p-Solutions to Singular Sublinear Elliptic Problems

Abstract

We solve the existence problem for the minimal positive solutions $u\in L^p(\Omega ,dx)$ to the Dirichlet problems for sublinear elliptic equations of the form \[ \begin{cases} Lu=σu^q+μ\qquad \quad \text{in} \quad Ω, \\ \liminf\limits_{x \rightarrow y}u(x) = 0 \qquad y \in \partial_{\infty}Ω, \end{cases} \] where $0<q<1$ and $Lu:=-\text{div}(\mathcal{A}(x)\nabla u)$ is a linear uniformly elliptic operator with bounded measurable coefficients. The coefficient $\sigma$ and data $\mu$ are nonnegative Radon measures on an arbitrary domain $\Omega \subset {\mathbb{R}}^n$ with a positive Green function associated with $L$. Our techniques are based on the use of sharp Green potential pointwise estimates, weighted norm inqualities, and norm estimates in terms of generalized energy.

arXiv comment12 pages

References in corpus (5)

• Nonlinear elliptic equations and intrinsic potentials of Wolff type
• A sublinear version of Schur's lemma and elliptic PDE
• Solutions to sublinear elliptic equations with finite generalized energy
• Finite energy solutions to inhomogeneous nonlinear elliptic equations with sub-natural growth terms
• Sublinear equations and Schur's test for integral operators

Cited by in corpus (1)

• Nonlocal Sublinear Elliptic Problems Involving Measures