Solvability of classes of semipositone problems.

摘要

We study nonnegative solutions for classes of nonlinear boundary value problems, referred in the literature as semipositone problems. These problems are motivated by various applications in the applied sciences and occur naturally as steady states of diffusion processes.; We first summarize the developments of semipositone problems to date. Most of the theory developed in the literature are for the single equation case. Here we provide three new results for semipositone systems. The first result provides the positivity of solutions for classes of semilinear elliptic systems in symmetric domains. The second result deals with the existence of a solution for semilinear elliptic systems in general bounded domains for classes of sublinear nonlinearities. Third result provides a solution for classes of quasilinear systems, including p-Laplacian systems, in annular regions for superlinear nonlinearities. Next, we establish two new results for semilinear elliptic single equations in general bounded regions. First we establish a multiplicity result for classes of sublinear nonlinearities. The second result establishes the instability of solutions for classes of non-monotone superlinear nonlinearities.; The positivity result is established by combining Maximum principle/refelection arguments and analysis of solutions near the boundary. Our existence results are established via sub-super solutions method and degree theory. The stability result employs the principle of linearized stability.