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 \[ \mathcal{L}u = σu^{q} + μ\quad \text{in} \;\; Ω, \] 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 $0<\gamma \le 1$, 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 $1<p\le 2$.

25 pages, published online in Calculus of Variations and Partial Differential Equations