Solutions to sublinear elliptic equations with finite generalized energy

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 $1<p\leq 2$.