Boundary Value Problems–Hamiltonian Systems

Summary

^*Having the general abstract theories developed in Chapter 4, in this chapter we deal with concrete boundary value problems (both for ordinary and partial differential equations). In the first section, we illustrate the uses of the so-called “variational method“, which is based on the critical point theory. We consider Dirichlet and Neumann problems for nonlinear problems driven by the p- Laplacian as well as semilinear periodic systems with indefinite linear part. We prove existence and multiplicity results. Then we illustrate the method of “upperlower solutions”. In fact combining this method with variational arguments, we prove three solutions theorems for p-Laplacian pde’s. We also deal with nonlinear nonvariational boundary value and provide at unifying framework that enables us to deal at the same time with the Dirichlet, Neumann, Sturm’Liouville, and periodic problems. Subsequently we illustrate the “degree-theoretical approach”, by proving multiplicity results for both ode’s and pde’s problems. In Section 5.4, we deal with perturbed elliptic eigenvalue problems and we prove existence and nonexistence results using Pohozaev identity. Then we prove maximum and comparison principles for the p-Laplacian. These results are useful tools when studying the existence of multiple nontrivial solutions. Finally, we examine Hamiltonian systems and deal with the minimal period and prescribed energy level problems.