Existence and Qualitative Properties of Solutions to Nonlinear Schrodinger-Poisson Systems

Our main equation of study is the nonlinear Schr¨odinger-Poisson system⇢−Du+u+r(x)fu = |u|p−1u, x 2 R3,−Df = r(x)u2, x 2 R3,with p 2 (2,5) and r : R3 ! R a nonnegative measurable function. In the spirit ofthe classical work of P. H. Rabinowitz [55] on nonlinear Schr¨odinger equations, wefirst prove existence of positive mountain-pass solutions and least energy solutions tothis system under different assumptions on r at infinity. Our results cover the rangep 2 (2,3) where the lack of compactness phenomena may be due to the combinedeffect of the invariance by translations of a ‘limiting problem’ at infinity and of thepossible unboundedness of the Palais-Smale sequences. In the case of a coercive r,namely r(x)!+• as |x|!+•, we then prove the existence of infinitely many distinctpairs of solutions. For p 2 (3,5) we exploit the symmetry of the problem bythe action of Z2 as well as some well-known properties of the Krasnoselskii-genus,whereas for p 2 (2,3] we use an appropriate abstract min-max scheme, which requiressome additional assumptions on r.After establishing these existence and multiplicity results, we are then interested inthe qualitative properties of solutions the singularly perturbed problem⇢−e2Du+lu+r(x)fu = |u|p−1u, x 2 R3−Df = r(x)u2, x 2 R3,with r : R3 ! R a nonnegative measurable function, l 2 R, and l > 0, taking advantageof a shrinking parameter e ⌧ 1. In particular, we seek to understand theconcentration phenomena purely driven by r. To this end, we first find necessaryconditions for concentration at points to occur for solutions in various functionalsettings which are suitable for both variational and perturbation methods. We thendiscuss a variational/penalisation method, which has been exploited in the case ofnonlinear Schr¨odinger equations, and discuss its applications to the present nonlinearSchr¨odinger-Poisson context, in the attempt of showing that the necessary conditionsare, in fact, sufficient conditions on r for point concentration of solutions. Finally,we present some preliminary results in this direction that elicit interesting standalonequalitative properties of the solutions.

Differentiation of the Choquet Integral and Its Application to Long-term Debt Ratings

Differentiation of the Choquet integral of a nonnegative measurable function with respect to a fuzzy measure on fuzzy measure space is proposed and it is applied to the capital investment decision making problem by Kaino and Hirota. In this paper, differentiation of the Choquet integral of a nonnegative measurable function is extended to differentiation of the Sipos Choquet integral of a measurable function and its properties will be discussed. First, the real interval limited Schmeidler Choquet integral and Sipos Choquet integral are defined for preparation, then the upper differential coefficient, the lower differential coefficient, the differential coefficient, and the derived function of the Choquet integral along the range of an integrated function are defined by the limitation process of the interval limited Choquet integral. Two examples are given, where the measurable functions are either a simple function or a triangular function. Basic properties of differentiation about swn and multiple with constant, addition, subtraction, multiplication, and division are shown. Then, the Choquet integral is applied to long-term debt ratings model, where the input is qualitative and quantitative data of corporations, and the output is Moody’s long-term debt ratings. The fuzzy measure, which is given as the importance of each qualitative and quantitative data, is derived from a neural net method. Moreover, differentiation of the Choquet integral is applied to the long-term debt ratings, where this differentiation indicates how much evaluation of each specification influence to the rating of the corporation.

A global estimate for the gradient in a singular elliptic boundary value problem

SynopsisWe investigate the singular problemwhere Ω is a bounded smooth domain, k a bounded, nonnegative measurable function and v Ω 0. For the solution u to this problem, which is shown to exist if k(x) > 0 on some subset of Ω with positive measure, a uniform bound for |∇u| in Ω is derived when k(x) ≧ ψ (dist (x, ∂Ω)) with ψ (s)/sv ∈ Lp(0, a) for some a > 0, p > 1.