Discontinuous semilinear problems in vector-valid function spaces
The existence of solutions to the discontinuous semilinear problem in vector-valued function spaces: \[ a(u, v)- \langle f, v\rangle_V+ \int_\Omega j^0(u, v) d\Omega\geq 0,\quad \forall v\in V \] is studied. The notation \(j^0(\cdot, \cdot)\) stands for the Clark's directional differential which satisfies the directional growth condition \((*)\) \(j^0(\xi, \eta- \xi)\leq \alpha(r) (1+ |\xi|^s)\). In the case \(1\leq s< 2\), the existence of solutions is established based on the finite-dimensional space regularization, Brouwer's fixed-point theorem and the Dunford-Pettis compactness criterion in \(L^1\). For \(s\geq 2\), some further hypotheses strengthening \((*)\) are imposed to guarantee that the solution can arise as a critical point of a locally Lipschitz functional. Then a nonsmooth minimax method in critical point theory due to K. C. Chang is applied to obtain the existence result in this case.
scientific article; zbMATH DE number 856939