Optimal Global And Boundary Asymptotic Behavior Of Large Solutions To The P-Laplacian Elliptic Equation

This paper is concerned with boundary and global asymptotic behavior of large solutions to the p-Laplacian elliptic equationΔpu(x)=b(x)f(u(x)),x∈Ω,where Ω is a bounded smooth domain in RN with N≥2,Δp is p Laplace operator,Δpu(x)=div(|▽u|p-2▽u),p>1,f∈C[0,∞)∩C1(0,∞)(or f∈C1(R)),which is increasing in[0,∞)(or R)and satisfies the Keller-Osserman type condition,b∈C(Ω)is nonnegative in Ω.The first chapter lists some basic results of the classical modelΔu=b(x)f(u),x∈Ω,u|(?)Ω=+∞,where Δ is Laplace operator,and then gives the symbols and three types basic func-tions involved in this article,and introduces the research background of the above-mentioned p-Laplacian elliptic equation.Subsequently,we find new structure condi-tions on f which plays a crucial role in both boundary and global asymptotic behavior of such solutions.The second chapter introduces some preliminaries,including the Karamata regu-lar variation theory,the weak comparison principle,the generalized Keller-Osserman condition and so on.In the third chapter,we analyzed the boundary asymptotic behavior of large solutions to the above-mentioned p-Laplacian elliptic equation by applying the per-turbation method,the weak comparison principle and the Karamata regular variation theory.In the fourth chapter,we studied the global asymptotic behavior of large solu-tions to the above-mentioned p-Laplacian elliptic equation by the sub supersolution method and the Karamata regular variation theory.We also reveal the optimal global asymptotic behavior of such solutions when the parameter on the weight function b tend to the corresponding critical value-p.In addition,when f does not satisfy Keller-Osserman condition,Ω is a ball and b has a high boundary singularity,we sup-ply a necessary and sufficient condition on b,and the existence of radially symmetric positive solutions to the above-mentioned p-Laplacian elliptic equation is proved by using the monotonic iteration method and the Arzela-Ascoli theorem.