Three solutions for a nonlocal Dirichlet boundary value problem involving the p(x)-Laplacian

Abstract

In this article, we consider a nonlocal problem involving the p(x)-Laplacian of the type

Applying a three critical points theorem from Bonanno [A critical points theorem and nonlinear differential problems, J. Global Optim. 28 (2004), pp. 249–258], we obtain the existence of two intervals of positive real parameters λ for which the above problem admits three weak solutions in , whose norms are uniformly bounded with respect to λ belonging to one of the two open intervals. In particular, we also prove some properties of p(x)-Kirchhoff–Laplace operator.

Keywords:

• p(x)-Laplacian
• nonlocal problem
• three critical points theorem

AMS Subject Classifications::

• 35D05
• 35J20
• 35J70

Acknowledgements

I wish to express my gratitude to the referee for reading this article carefully and making several corrections and remarks. This research was supported by the NNSFC (10971087), NWNU-LKQN-10-21.