Elliptic equations involving general subcritical source nonlinearity and measures

Abstract

In this article, we study the existence of positive solutions to elliptic equation (E1) $$(-\Delta {)}^{\alpha }u=g(u)+\sigma \nu \mathrm{in}\Omega ,$$ subject to the condition (E2) $$u=\varrho \mu \mathrm{on}\partial \Omega \mathrm{if}\alpha =1\mathrm{or}\mathrm{in}{\Omega }^c\mathrm{if}\alpha \in (0,1),$$ where $\sigma ,\varrho \ge 0$, $\Omega$ 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 $\sigma ,\varrho \ge 0$ small enough.

References in corpus (1)

• Existence and stability of solutions of general semilinear elliptic equations with measure data

Cited by in corpus (2)

• Existence, Non-existence, Uniqueness of solutions for semilinear elliptic equations involving measures concentrated on boundary
• Boundary value problem with measures for fractional elliptic equations involving source nonlinearities