Nonlinear nonlocal equations involving subcritical or power nonlinearities and measure data

Konstantinos T. Gkikas; Konstantinos T. Gkikas

doi:10.3934/mine.2024003

Mathematics in Engineering

2024, Volume 6, Issue 1: 45-80. doi: 10.3934/mine.2024003

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Research article

Nonlinear nonlocal equations involving subcritical or power nonlinearities and measure data

Konstantinos T. Gkikas ^{1,2
,
,}

1.
Department of Mathematics, University of the Aegean, 83200 Karlovassi, Samos, Greece
2.
Department of Mathematics, National and Kapodistrian University of Athens, 15784 Athens, Greece

Received: 13 October 2023 Revised: 04 January 2024 Accepted: 09 January 2024 Published: 15 January 2024

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.
- fractional $ p $-Laplace operator,
- critical exponents,
- Bessel capacities,
- Wolff potentials
Citation: Konstantinos T. Gkikas. Nonlinear nonlocal equations involving subcritical or power nonlinearities and measure data[J]. Mathematics in Engineering, 2024, 6(1): 45-80. doi: 10.3934/mine.2024003

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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.

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