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We study the uniqueness and nondegeneracy of positive solutions of div (ρ∇u) + ρ(−gu + hu p ) = 0 in a ball, the entire space, an annulus, or an exterior domain under the Dirichlet boundary condition. Mathematics Subject Classification

The knapsack problem and the minimum spanning tree problem are both fundamental in operations research and computer science. We are concerned with a combination of these two problems. That is, we are given a knapsack of a fixed capacity, as well as an undirected graph where each edge is associated with profit and weight. The problem is to fill the knapsack with a feasible spanning tree such that the tree profit is maximized. We prove this problem NP-hard, present upper and lower bounds, develop a branch-and-bound algorithm to solve the problem to optimality and propose a shooting method to accelerate computation. We evaluate the developed algorithm through a series of numerical experiments for various types of test problems.

Planar elastica problem is a classical but has broad connections with various fields, such as elliptic functions, di¤erential geometry, soliton theory, material mechanics, etc. This paper regards classical elastica as a theory corresponding to Lebesgue L 2 case, and extends it to L p cases. For the sake of the e¤ect of p-Laplacian, novel curious solutions appear especially for cases p > 2. These solutions never appear in 1 < p a 2 cases and we call them flat-core solutions according to Takeuchi [6, 7].

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In order to study the buckled states of an elastic ring under uniform pressure, Tadjbakhsh and Odeh [14] introduced an energy functional which is a linear combination of the total squared curvature (elastic energy) and the area enclosed by the ring. We prove that the minimizer of the functional is not a disk when the pressure is large, and its curvature can be expressed by Jacobian elliptic cn( • ) function. Moreover, the uniqueness of the minimizer is proven for certain range of the pressure.

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Abstract. A discrete version of the Sobolev inequalty in the Hilbert spacewhich is equipped with a suitable inner product, is derived. The best constant and best function of the discrete Sobolev inequality are also obtained from the theory of reproducing kernels, and are expressed by means of discrete analogues of the wellknown Bernoulli polynomials. Some interesting properties of these discrete Bernoulli polynomials are also discussed.