A Priori Estimate And The Existence Of Positive Solution For Semilinear Elliptical Equations On The Robin Problem

In this paper, we are concerned with the question of existence of a pair of smooth functions u(x) and v(x) satisfying the following system: Whereα,β, p , q> 0are positive constants, 1/(p + 1) + 1/(q + 1)≥(n-1)/n, (n|-) is the outward unit normal on (?)Ω,Ω∈Rⁿ is a smooth bounded domain.We firstly establish priori estimates for pq > 1and pq < 1; After that we prove the existence result: The system has a classical positive solution (u, v)∈C^{2, r}(Ω|—)×C^2,r(Ω|—), supposing p ,q satisfies some conditions; At the end we have some useful theorems for general semi-linear elliptic system.