Nonlocal Sublinear Elliptic Problems Involving Measures

Abstract

We study Dirichlet problems for fractional Laplace equations of the form $(-\Delta)^{\frac{\alpha}{2}} u = f(x,u)$ in $\mathbb{R}^{n}$ for $0<\alpha<n$ where the nonlinearity $f(x,u) = \sum_{i=1}^{M} \sigma_{i} u^{q_i} + \omega$ involves sublinear terms with $0<q_{i}<1$ and the coefficients $\sigma_{i}, \omega$ are nonnegative locally finite Borel measures on $\mathbb{R}^n$. We develop a potential theoretic approach for the existence of positive minimal solutions in Lorentz spaces to the problems under certain assumptions on $\sigma_{i}$ and $\omega$. The uniqueness properties of such solutions are discussed. Our techniques are also applicable to similar sublinear problems on uniform bounded domains when $0<\alpha< 2$, or on arbitrary domains with positive Green's functions in the classical case $\alpha =2$.