A quasi-linear nonlocal Venttsel' problem of Ambrosetti–Prodi type on fractal domains

Maria Rosaria Lancia; Alejandro Vélez-Santiago; Paola Vernole

doi:10.3934/dcds.2019184

Article Contents

2019, Volume 39, Issue 8: 4487-4518. Doi: 10.3934/dcds.2019184

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A quasi-linear nonlocal Venttsel' problem of Ambrosetti–Prodi type on fractal domains

1.
Dipartimento di Scienze di Base e Applicate per I'Ingegneria, Sapienza Università di Roma Roma, Italy, Via A. Scarpa 16, 00161 Roma, Italy
2.
Department of Mathematical Sciences, University of Puerto Rico at Mayagüez, Mayagüez, Puerto Rico, 00681, USA

^* Corresponding author: A. Vélez-Santiago
^* Corresponding author: A. Vélez-Santiago

Received: July 31, 2018

Published: May 2019

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Abstract

We investigate the solvability of the Ambrosetti-Prodi problem for the p-Laplace operator ∆p with Venttsel' boundary conditions on a twodimensional open bounded set with Koch-type boundary, and on an open bounded three-dimensional cylinder with Koch-type fractal boundary. Using a priori estimates, regularity theory and a sub-supersolution method, we obtain a necessary condition for the non-existence of solutions (in the weak sense), and the existence of at least one globally bounded weak solution. Moreover, under additional conditions, we apply the Leray-Schauder degree theory to obtain results about multiplicity of weak solutions.

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Mathematics Subject Classification: 35J62, 35J92, 35D30, 35B45, 35B65.

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Figure 1. Pre-fractal Koch snowflake

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