Existence Of Solutions To Semilinear Elliptic Equations In Exterior Domains

In this paper, the following two kinds of semilinear elliptic partial differential equations will be considered:whereΩ(?) R~n(n≥3) is an exterior domain, which has a closed interior boundary satisfying the exterior sphere condition, and c(x, s)≤0.whereΩ(?) R~n(n≥3) is an exterior domain, 2 < q < 2~*, 2~* = 2n/n-2,μ≥0.Under a set of suitable hypotheses, with the aid of the super-subsolution method and the mountain pass lemma, the existence of solutions for these two kinds of semilinear elliptic equations is given.The paper is mainly divided into four sections.The first section is an introduction of the whole paper. We talk about the background of this paper, and the plans for the research of the problem.The next section consists of some basic definitions and theorems, including the maximum, Holder continuity, mountain pass lemma, and so on. They are the foundations and tools of the later work.The third section is the main part of this paper, where we discuss the exsistence of solutions by using the super-subsolution theorem and the mountain pass lemma, and we give the sufficient conditions for the existence of solutions.At last, we summarize the work of whole paper and point out some open problems.