Existence And Uniqueness Of Positive Solutions For The Logistic Type Elliptic Equation

In this paper,we mainly apply the method of upper and lower solutions and the comparison principle to study the following Logistic type elliptic equation:where ?(?)RN is a bounded domain with C2 boundary(?)Q,bi(x)(i=1,2)are non-negative continuous function over ?,pi are constants satisfying pi>1(i=1,2)and ??R1 is a parameter.When ? and bi(x)(i=1,2)satisfy some conditions,we investigate the existence and uniqueness of positive solutions for the above equation in the non-degenerate case(i.e.,b1(x)+b2(x)>0,(?)x??)and the degeneration case(i.e.,b1(x)+b2(x)(?)0 and {x??:b1(x)+b2(x)=0} is a sub-domain).In order to better understand the properties of the solutions of the degenerate Logistic equation,it is necessary to further study the related boundary blow-up problem,and for the more general case,we consider the following problem:where the boundary(?)? is divided into two parts,?? is a nonempty open and closed subset of(?)?,?0=(?)?\??(which might be empty),a is a continuous function on ?.Under the rather general conditions on coefficient,we obtain the uniqueness and the asymptotic behavior of boundary blow-up solutions of the equation above.Our results extend relevant works of[24].