In this paper we study a Dirichlet problem of a system involving fractional (p, q)-Laplacian operators in bounded regular domains of RN and a real parameter λ. Since the involved fractional Laplacian operators can have different order, the classical definitions of the Nehari manifold for systems and of the Fibering mapping are not suitable. In this paper, we modify these definitions to solve the Dirichlet problem under consideration. Then, by virtue of the properties of the first eigenvalue λ1 for a related system, we prove that there exists a positive solution for the problem when λ<λ1 by the modified definitions. Moreover, we obtain the bifurcation property when λ → λ1-. Finally, thanks to the Picone identity, we also obtain a nonexistence result when λ≥λ1.
Existence of Nonnegative Solutions for a Class of Systems Involving Fractional (p,q)-Laplacian Operators
Patrizia Pucci
2018
Abstract
In this paper we study a Dirichlet problem of a system involving fractional (p, q)-Laplacian operators in bounded regular domains of RN and a real parameter λ. Since the involved fractional Laplacian operators can have different order, the classical definitions of the Nehari manifold for systems and of the Fibering mapping are not suitable. In this paper, we modify these definitions to solve the Dirichlet problem under consideration. Then, by virtue of the properties of the first eigenvalue λ1 for a related system, we prove that there exists a positive solution for the problem when λ<λ1 by the modified definitions. Moreover, we obtain the bifurcation property when λ → λ1-. Finally, thanks to the Picone identity, we also obtain a nonexistence result when λ≥λ1.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
