几类具Pucci算子的椭圆方程组解的存在性

Existence of Some Elliptic Systems Involving the Pucci Operator

研究完全非线性椭圆方程组解的存在性问题,其中ΩR^n,n≥2是有界光滑区域,—Μ_(λ,Λ)^+为具参数0<λ≤Λ的Pucci算子.首先,对f_i,i=1,2为一致有界函数的情形,证明了此方程组存在有界非负解.其次,当{f_1,f_2}是拟增的,且方程组存在有序上、下解时,利用上、下解方法,并结合增算子的不动点定理证明了此方程组存在最大非负解和最小非负解.当{f_1,f_2}是拟减或混拟单调时,使用Schauder不动点定理证明了此方程组至少存在一个非负解.针对此方程组中f_i,i=1,2的某些特殊形式,证明了相应方程组正解的存在性.最后给出了应用实例.

We study the existence of solutions for the fully nonlinear elliptic system {--Μλ,Λ^＋（D^2ui）=fi（x,u1,u2）,x∈Ω ui≥0,x∈Ω,i=1,2 ui=0 ,x∈Ω whereΩRn,n≥2 is a smooth bounded domain,-Mλ,∧^＋is the Pucci operator with parameters 0 λ≤A.Firstly,we show that the system has a bounded nonneg- ative solution if fi,i = 1,2 are uniformly bounded functions.Secondly,we prove that the system has a maximal nonnegative solution and a minimal nonnegative solution by using the method of super-subsolution combining with a fixed point theorem of increasing operator if {f1,f2} is quasi-increasing and the system has a supersolution and a subsolution.If {f1,f2} is either quasi-decreasing or blended quasi-monotony,the existence of a non-negative solution is presented by the Schauder fixed point theorem. For some particular type of fi,the existence of positive solutions for the system is also obtained.We finally give some examples as application.