UNIQUENESS OF NONTRIVIAL SOLUTIONS FOR DEGENERATE MONGE-AMPERE EQUATIONS

In this paper, we are interested in the following degenerate elliptic Monge--Ampere equation: { detD(2)u=f( - u) in Omega u= 0 on partial derivative Omega A typical example in the present paper is f(t) =lambda t(n)+Lambda t(q),q not equal n, lambda >= 0. When f(t) = (lambda (nq)t)(nq), (1.1) changes to{ detD(2)u=f( -Lambda(nq) u)(nq) in Omega , u= 0 on partial derivative Omega. For q= 1, (1.2) is the eigenvalue problem of the Monge--Ampere equation. The eigen-value problem was first studied by Lions [16] in bounded smooth uniformly convex domain \Omega . The uniqueness of the eigenvalue andC1,1(Omega ) regularity of the eigen functions were also proved in [16]. For general q >0 and Omega bounded smooth uniformly convex, Tso first proved the existence of nontrivial solutions u is an element of C infinity (Omega )boolean AND C(Omega in[23]. Hartenstine [10] extended Tso's results to strictly convex domain for (1.2) and q is an element of (0,1). Recently, Le [14] got the corresponding existence results in general convex domain for (1.2) and q is an element of (0,+infinity). It should be pointed out that Tso's results also hold true for general f(t) with suitable structural conditions.