In this paper we consider the existence of ground states for some 2- coupled nonlinear Schrödinger systems with or without potentials. Under various conditions on the parameters in the equations, we prove the existence of ground states.

In this note, we investigate the existence of the minimal solution and the uniqueness of the weak extremal (probably singular) solution to the biharmonic equation Δ^2ω=λg(ω) with Dirichlet boundary condition in the unit ball in R^n, where the source term is logarithmically convex. An example is also given to illustrate that the logarithmical convexity is not a necessary condition to ensure the uniqueness of the extremal solution.

Results concerning existence, uniqueness and local regularity of Inhomogeneous Schrödinger flow from T^d_R into S^2, are presented by using the difference method, where T^d_R=R^d/(R·Z)^d.

We consider the problem of whether the equation $Δu = p(x) f (u)$ on $R^N, N ≥ 3$, has a positive solution for which $lim_{|x|→∞} u(x)=∞$ where f is locally Lipschitz continuous, positive, and nondecreasing on (0,∞) and satisfies $∫^∞_1[F(t)]^{-1/2}dt=∞$ where $F(t)=∫^t_0f(s)ds$. The nonnegative function p is assumed to be asymptotically radial in a certain sense. We show that a sufficient condition to ensure such a solution u exists is that p satisfies $∫^∞_0r\min_{|x|=r}p(x)dr=∞$. Conversely, we show that a necessary condition for the solution to exist is that p satisfies $∫^∞_0r^{1+ε}\min_{|x|=r}p(x)dr=∞$ for all $ε > 0$.

Lagrange's variation-of-constantsmethod for solving linear inhomogeneous ordinary differential equations (ode's) is replaced by amethod based on the Loewy decomposition of the corresponding homogeneous equation. It uses only properties of the equations and not of its solutions. As a consequence it has the advantage that it may be generalized for partial differential equations (pde's). It is applied to equations of second order in two independent variables, and to a certain system of third-order pde's. Therewith all possible linear inhomogeneous pde's are covered that may occur when third-order linear homogeneous pde's in two independent variables are solved.

We study logarithmic gradient estimate to smooth admissible solutions of the Hessian-type equations on S^n. The equations contain a Monge-Ampère equation arising in designing a reflecting surface in geometric optics as a special case.