Existence And Multiplicity Of Positive Solutions For A Class Of Elliptic Systems With Perturbations

In this paper, we study the following elliptic systemswhereΩis a bounded smooth in R~N; F∈C~1((?)×(R~+)~2, R~+), here, R~+:= [0, +∞); g, he C~1(?)\{0};ε>0 is a parameter.We find that the existence of positive solutions for problem (1) is closely related to the nonnegative solutions of the following two elliptic problemsandTherefore, we consider the existence and multiplicities of positive solutions for problem (1) under this conditions which problem (2) and problem (3) have nonnegative solutions.Firstly, using sub-supersolution method and the maximum principle, the existence of positive solutions for problem (1) is proved by assume that F satisfies super-linear conditions, (?) (x,z),(?)(x,z)∈C((?)×(R~+)~2,R~+) are strictly increasing function about u and v in R~+\{0};|▽F(x,z)|=o(|z|) as (|z|→0) uniformly in x∈Ω. Similarly, assume that F satisfies lower-linear conditions, and F(x,z) = F(z) is homogeneous of degreeμ(μ∈(1,2)), that is, F(tz) = t~μF(z) for all t>0, we prove that problem (1) have positive solutions. Moreover, for the supercritical and loer-linear conditions, problem (2) and problem (3) have nonnegative solutions are a necessary condition that guarantees the existence of positive solutions for problem (1).Secondly, for the case of super-linear conditions, the multiplicities of positive solutions for problem (1) in subcritical and critical conditions are discussed. By means of the variational approach, assume that F satisfies (?)(x,z), (?)(x,z)∈C((?)×(R~+)~2,R~+) are strictly increasing function about u and v in R~+\{0};|▽F(x,z)|=o(|z|) as (|z|→0) uniformly in x∈Ω; there exists a number r satisfying 22 such that z·▽F(x, z)≥qF(x, z) for all (x, z)∈(?)×(R~+)~2 ; F(u, 0)=F(0, v)=(?)(0, v)=(?)(u,0)=0, where u,v∈R~+, we prove that the multiplicities of positive solutions for problem (1). Assume that F satisfies critical conditions: F(x,z) = F(z) is homogeneous of degree 2~* and F(u,0) = F(0,v)=(?)(0,v) = (?)(u,0)=0, where u, v∈R~+, the multiplicities of positive solutions for problem (1) are proved by means of the variational approach.Finally, we generalize problem (1) to quasi-linear elliptic systems and prove the existence of nonnegative solutions under suitable conditions.