Nonlinear nonlocal equations involving subcritical or power nonlinearities and measure data

Abstract

Let $ s\in(0, 1), $ $ 1 < p < \frac{N}{s} $ and $ \Omega\subset{\mathbb R}^N $ be an open bounded set. In this work we study the existence of solutions to problems ($ E_\pm $) $ Lu\pm g(u) = \mu $ and $ u = 0 $ a.e. in $ {\mathbb R}^N\setminus \Omega, $ where $ g\in C({\mathbb R}) $ is a nondecreasing function, $ \mu $ is a bounded Radon measure on $ \Omega $ and $ L $ is an integro-differential operator with order of differentiability $ s\in(0, 1) $ and summability $ p\in(1, \frac{N}{s}). $ More precisely, $ L $ is a fractional $ p $-Laplace type operator. We establish sufficient conditions for the solvability of problems ($ E_\pm $). In the particular case $ g(t) = |t|^{ \kappa-1}t; $ $ \kappa > p-1, $ these conditions are expressed in terms of Bessel capacities.

Keywords

• fractional $ p $-laplace operator
• critical exponents
• bessel capacities
• wolff potentials