Bounded Solutions for Nonlocal Boundary Value Problems on Lipschitz Manifolds with Boundary

Abstract

Abstract We consider nonlinear nonlocal boundary value problems associated with fractional operators (including the fractional p-Laplace and the regional fractional p-Laplace operators) and subject to general (fractional-like) boundary conditions on bounded domains with Lipschitz boundary. Under suitable conditions on the nonlinearities of our system, we establish the existence of bounded solutions and provide explicit L ∞ ${L^{\infty}}$ -estimates of solutions which are optimal with respect to the inhomogeneous “sources” present in the system. As application, these results are shown to apply to a class of nonlinear nonlocal equations for the Dirichlet fractional p-Laplacian and regional fractional p-Laplace with a dissipative nonlinearity, and to a class of semilinear nonlocal boundary value problems with fractional Wentzell–Robin boundary conditions corresponding to the so-called fractional Wentzell Laplacian.