Solutions and multiple solutions for problems with the p-Laplacian
In this paper we consider two nonlinear elliptic problems driven by the p-Laplacian and having a nonsmooth potential (hemivariational inequalities). The first is an eigenvalue problem and we prove that if the parameter λ < λ2 = the second eigenvalue of the p-Laplacian, then there exists a nontrivial smooth solution. The second problem is resonant both near zero and near infinity for the principal eigenvalue of the p-Laplacian. For this problem we prove a multiplicity result. Our approach is variational based on the nonsmooth critical point theory.
First and second eigenvalues of the p-Laplacian, Linking sets, Local linking condition, Locally Lipschitz potential, Nonsmooth C and PS-conditions
Hu, Shouchuan, and Nikolaos S. Papageorgiou. "Solutions and multiple solutions for problems with the p-Laplacian." Monatshefte für Mathematik 150, no. 4 (2007): 309-326.
Monatshefte fur Mathematik