Bounded Positive Entire Solutions Of Semilinear Elliptic Partial Differential Equations

In this paper, a class of semilinear elliptic partial differential equations will be considered:△u + p(x)u^α + q(x)u^-β - h(x)u^γ = 0, x ∈ R^N,where α ∈ [0, l),β> 0,γ≥1 are constants. Under suitable hypotheses, the equation possesses a positive entire solution u(x) > 0 with θ ∈ (0,1).The paper is mainly divided into four chapters.The first section is the introduction of the whole paper. We talk about the background of this paper, and make plans for the research of the problem.Next section of the paper is composed of some basical definitions and theorems, including the maximum, Holder continuity, and so on. They are the foundations and tools of the later work, and we will use them without proof.The third section is the main part of this paper, where we discuss the existence of positive entire solutions by the super-subsolution method and some fixed point theorem. In this section, we prove the sufficient conditions for the existence of the solutions. Then, a necessary condition is mentioned.At last, we summarize the work of the whole paper and give some open problems about the equation.