We study the Dirichlet boundary value problem for the p-Laplacian of the form −∆pu − λ1|u| p−2u = f in Ω, u = 0 on ∂Ω, where Ω ⊂ N is a bounded domain with smooth boundary ∂Ω, N ≥ 1, p > 1, f ∈ C(Ω) and λ1 > 0 is the first eigenvalue of ∆p. We study the geometry of the energy functional Ep(u) = 1⁄ p ∫ Ω |∇u| p − λ1⁄ p ∫ Ω |u| p − ∫ Ω fu and show the difference between the case 1 <p< 2 and the case p > 2. We also give the characterization of the right hand sides f for which the above Dirichlet problem is solvable and has multiple solutions.
We study the question of the existence, uniqueness, and continuous dependence on parameters of the Carathéodory solutions to the Cauchy problem for linear partial functional-differential equations of hyperbolic type. A theorem on the Fredholm alternative is also proved. The results obtained are new even in the case of equations without argument deviations, because we do not suppose absolute continuity of the function the Cauchy problem is prescribed on, which is rather usual assumption in the existing literature.