Asymptotic Behaviour Of Solutions For Semilinear Elliptic Equations With Gradient Term

The existence of large solutions and entire bounded solutions inΩfor the semilinear elliptic equation with gradient termΔu + (?) u = p ( x ) uα+ q ( x )uβandΔu + (?)u = p ( x ) f (u ) + q ( x )g (u ) are discussed in this paper, whereΩ(?) R N is either a smooth bounded domain or the whole space. The paper is divided into four sections.In the preface we state the background of this paper and the newest results.Chapter one we introduce some fundamental knowledge, which include the H(?)lder continuity, maximum principle, the Schauder interior estimates, Arzela-Ascoli theorem, and so on. These theorems are important tools for the later work. We will directely use them and no longer prove them in this paper.Chapter two we mainly discuss the existence of large solutions and entire bounded solutions for the semilinear elliptic equation with gradient termΔu + (?) u = p ( x ) uα+ q ( x )uβ. By using the super-subsolution method, the perturbed method and the inner estimation theorem, we obtain the necessary and sufficient conditions for the existence of entire large solutions of the equations and the sufficient conditions for the existence of entire bounded solutions of the equation in R n respectively. Moreover, the nonexistence of large solutions of the equations in a bounded domainΩwith smooth boundary in R n is proved. At last, we extend the above results to the generally semilinear elliptic equationΔu + (?) u = p ( x ) f (u )+ q ( x ) g ( u ) and investigate the existence of large solutions and entire bounded solutions.Chapter three we give three examples to show the influence of functions p ( x ),q ( x ) on the existence of large solutions for the equations. The examples indicate the practical significance of the research problems.