On a class of hemivariational inequalities at resonance.
The paper studies a Dirichlet boundary value problem involving the \(p\)-Laplacian \(\Delta_p\) and a locally Lipschitz potential which is resonant at the first eigenvalue of \(-\Delta_p\). This type of problems is called resonant hemivariational inequalities. A specific feature of the paper is that it is only assumed a unilateral growth condition for the generalized directional derivative of the locally Lipschitz potential. The main result of the paper establishes the existence of a solution in an appropriate weak sense. The approach is variational, but not relying directly on the nonsmooth critical point theory. Specifically, due to the generality of the imposed growth condition, the functional associated to the problem is not locally Lipschitz, so it is not possible to apply readily the critical point theory for locally Lipschitz functionals. To overcome this difficulty the authors develop a finite-dimensional approximation, apply on each finite dimensional space a version of the mountain pass theorem for locally Lipschitz functions to obtain approximate solutions and pass to the limit using the Dunford-Pettis criterion.
scientific article; zbMATH DE number 2057710
scientific article