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

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.