Elliptic equations involving general subcritical source nonlinearity and measures

In this article, we study the existence of positive solutions to elliptic equation (E1) $$(-\mathrm{\Delta }{)}^{\alpha }u=g(u)+\sigma \nu \mathrm{i}\mathrm{n}\mathrm{\Omega },$$ subject to the condition (E2) $$u=\varrho \mu \mathrm{o}\mathrm{n}\mathrm{\partial }\mathrm{\Omega }\mathrm{i}\mathrm{f}\alpha =1\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{n}{\mathrm{\Omega }}^c\mathrm{i}\mathrm{f}\alpha \in (0,1),$$ where $\sigma ,\varrho \ge 0$, $\mathrm{\Omega }$ is an open bounded $C^2$ domain in ${\mathbb{R}}^N$, $(-\mathrm{\Delta }{)}^{\alpha }$ denotes the fractional Laplacian with $\alpha \in (0,1)$ or Laplacian operator if $\alpha =1$, $\nu ,\mu$ are suitable Radon measures and $g:{\mathbb{R}}_{+}\mapsto {\mathbb{R}}_{+}$ is a continuous function. We introduce an approach to obtain weak solutions for problem (E1)-(E2) when $g$ is integral subcritical and $\sigma ,\varrho \ge 0$ small enough.