Elliptic equations involving general subcritical source nonlinearity and measures

In this article, we study the existence of positive solutions to elliptic equation (E1) $$(-\Delta)^\alpha u=g(u)+\sigma\nu \quad{\rm in}\quad \Omega,$$ subject to the condition (E2) $$u=\varrho\mu\quad {\rm on}\quad \partial\Omega\ \ {\rm if}\ \alpha=1\qquad {\rm or\ \ in}\ \ \Omega^c \ \ {\rm if}\ \alpha\in(0,1),$$ where $\sigma,\varrho\ge0$, $\Omega$ is an open bounded $C^2$ domain in $\mathbb{R}^N$, $(-\Delta)^\alpha$ 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 $\sigma,\varrho\ge0$ small enough.