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 $R^n$ 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 $R^n$. 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 ω. 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 $\alpha =2$.