Nonlocal Sublinear Elliptic Problems Involving Measures

TL;DR

This paper investigates the existence and uniqueness of positive solutions to fractional Laplace equations with measure data and sublinear nonlinearities, using potential theory and Lorentz space techniques.

Contribution

It introduces a potential theoretic framework for solving fractional elliptic problems with measure coefficients and sublinear terms, extending to various domain types and fractional orders.

Findings

Existence of positive minimal solutions under certain measure conditions.

Uniqueness properties of solutions are established.

Applicable to bounded and unbounded domains with Green's functions.

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 $\alpha =2$.