Solutions to sublinear elliptic equations with finite generalized energy

TL;DR

This paper establishes necessary and sufficient conditions for the existence of finite energy positive solutions to sublinear elliptic equations with measure data, expanding understanding of solutions in generalized energy spaces.

Contribution

It provides new criteria for solutions with finite generalized energy in sublinear elliptic equations involving measures and variable coefficients.

Findings

Characterization of solutions with finite generalized energy.

Conditions for existence of solutions in Dirichlet space.

Extension to equations with measure data and variable coefficients.

Abstract

We give necessary and sufficient conditions for the existence of a positive solution with zero boundary values to the elliptic equation \[ \mathcal{L}u = \sigma u^{q} + \mu \quad \text{in} \;\; \Omega, \] in the sublinear case $0<q<1$, with finite generalized energy: $\mathbb{E}_{\gamma}[u]:=\int_{\Omega} |\nabla u|^{2} u^{\gamma-1}dx<\infty$, for $\gamma >0$. In this case $u \in L^{\gamma+q}(\Omega, \sigma)\cap L^{\gamma}(\Omega, \mu)$, where $\gamma=1$ corresponds to finite energy solutions. Here $\mathcal{L} u:= -\,\text{div}(\mathcal{A}\nabla u)$ is a linear uniformly elliptic operator with bounded measurable coefficients, and $\sigma$, $\mu$ are nonnegative functions (or Radon measures), on an arbitrary domain $\Omega\subseteq \mathbb{R}^n$ which possesses a positive Green function associated with $\mathcal{L}$. When $0<\gamma\leq 1$, this result yields sufficient conditions for the existence of a positive solution to the above problem which belongs to the Dirichlet space $\dot{W}_{0}^{1,p}(\Omega)$ for $1<p\leq 2$.