In this paper we study the existence of positive solutions for nonlinear elliptic problems driven by the $p$-Laplacian differential operator and with a nonsmooth potential (hemivariational inequalities). The hypotheses, in the case $p=2$ (semilinear problems), incorporate in our framework of analysis the so-called asymptotically linear problems. The approach is degree theoretic based on the fixed-point index for nonconvex-valued multifunctions due to Bader .
"Degree theoretic methods in the study of positive solutions for nonlinear hemivariational inequalities." Differential Integral Equations 19 (2) 223 - 240, 2006. https://doi.org/10.57262/die/1356050526