Boundary Value Problems
We consider a nonlinear elliptic equation driven by the p-Laplacian with a nonsmooth potential (hemivariational inequality) and Dirichlet boundary condition. Using a variational approach based on nonsmooth critical point theory together with the method of upper and lower solutions, we prove the existence of at least three nontrivial smooth solutions: one positive, the second negative, and the third sign changing (nodal solution). Our hypotheses onthe nonsmooth potential incorporate in our framework of analysis the so-called asymptotically p-linear problems.
Agarwal, R. P., Filippakis, M. E., O'Regan, D., & Papageorgiou, N. S. (2009). Constant sign and nodal solutions for problems with the p-laplacian and a nonsmooth potential using variational techniques. Boundary Value Problems, 2009, 1-32.