Solutions of Elliptic Equations ΔU+K(x)e2u=f(x) (Journal of Partial Differential Equations, 1991)
In this paper we consider the elliptic equation Δu + K(x)e^{2u} = f(x), which arises from prescribed curvature problem in Riemannian geometry. It is proved that if K(x) is negative and continuous in R², then for any f ∈ L²_{loc} (R²) such that f(x) ≤ K(x), the equation possesses a positive solution. A uniqueness theorem is also given.
Keywords: elliptic equations; prescribed curvature problem; monotone operators; Kato's inequality