摘要(英) |
In the last two decades, significant progress has been made in the theory of nonlinear systems of partial differential equations. There has been much effort dedicated to the developments, motivated by applications to the natural sciences such as physics, chemistry and biology. There is also a substantial amount of consequences related to topical issues for the study of medical and ecological sciences, and other applications involving fluid, plasma, reactor dynamics and so on. In particular, systems of elliptic partial differential equations are mainly encountered in stationary problems of the theory of heat and mass transfer in reacting media, the theory of chemical reactors, combustion theory, mathematical biology and biophysics etc.
This dissertation, consisting of four main parts, is devoted to studying some qualitative properties of solutions, such as the existence, uniqueness and structure of solutions to four specific kinds of nonlinear elliptic systems. In Part 1, sublinear elliptic systems are considered, and the existence, uniqueness and stability of solutions are derived via the bifurcation theory and monotonicity method. Part 2 deals with a cooperative Hamiltonian system, which can also be viewed as a natural extension of the single equation arising from investigating the stationary states of the nonlinear Schrödinger equation. By applying linearization techniques and the implicit function theorem, a complete structure of solutions is clarified. Part 3 studies a Liouville-type system with singularity at the origin. The existence and uniqueness of solutions to the Dirichlet boundary value problem are established. In addition, the structure of solutions in terms of some specific quantities, which are viewed as the total curvature or flux in the single case, will be provided as well. Finally, a singular Bennett-type system coming from modeling dissipative stationary plasmas is introduced in Part 4, and its blow-up phenomena will be investigated.
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