asymptotic behavior of positive solutions of a semilinear dirichlet problem abstract in this paper we study the asymptotic behavior of the unique positive classical solution to the following semilinear boundary value problem u a x u 0 x u 0 in u 0 here is a bounded c 1 1 domain 1 and the function a is in c l o c 0 1 such that there exists c 0 satisfying for each x 1 c a x x exp x z t t d t c where 2 d d i a m x d i s t x and z is a continuous function on 0 with z 0 0

on a semilinear fractional dirichlet problem on a bounded domain let d be a bounded c 1 1 domain in r d d 2 0 a 2 and s 1 we are concerned with the existence the uniqueness and the asymptotic behavior of positive continuous solution for the following semilinear fractional differential equation email protected d a 2u a x u s x ind with the boundary dirichlet email protected email protected email protected x 2 au x 0 here email protected d a 2 is the fractional laplacian associated to the subordinate killed brownian motion process in d and d x d x d denotes the euclidean distance between x and d our arguments are based on potential theory associated to the fractional laplacian and some karamata regular variation theory tools

on the existence of positive solutions of singular nonlinear elliptic equations with dirichlet boundary conditions abstract this paper is concerned with the existence of positive solutions of the singular nonlinear elliptic equation with a dirichlet boundary condition u f x u in u on where f is a borel measurable function in 0 such that f x u v x u for some 0 and v satisfying some appropriate conditions in particular we show that the above problem has positive solutions whenever inf is greater than a positive quantity given by and v

estimates on green functions and poisson kernels for symmetric stable processes one of the most basic and most important subfamily of levy processes is symmetric stable processes a symmetric stable process x on rn is a levy process whose transition density p t x y relative to the lebesgue measure is uniquely determined by its fourier transform rn e ix p t x dx e t here must be in the interval 0 2 when 2 we get a brownian motion running with a time clock twice as fast as the standard one brownian motion plays a central role in modern probability theory and has numerous important applications in other scientific areas as well as in many other branches of mathematics thus it has been intensively studied in this paper symmetric stable processes are referred to the case when 0 2 unless otherwise specified in the last few years there has been an explosive growth in the study of physical and economic systems that can be successfully modeled with the use of stable processes stable processes are now widely used in physics operations research queuing theory mathematical finance and risk estimation in some physics literatures symmetric stable processes are called levy flights and they have been applied to a wide range of very complex physics issues such as turbulent diffusion vortex dynamics anomalous diffusion in rotating flows and molecular spectral fluctuations in mathematical finance stable processes can be used to model stock returns in incomplete market for these and more applications of stable processes please see the interesting book 14 by janicki and weron and the references therein and the recent article 15 by klafter shlesinger and zuomofen in order to make precise predictions about natural phenomena and to better cope with these

existence results for some polyharmonic problems in the half space abstract we study the existence of positive solutions of the m polyharmonic nonlinear elliptic equation m u f u 0 in the half space r n x x 1 x n r n x n 0 n 2 and m 1 our purpose is to give two existence results for the above equation subject to some boundary conditions where the nonlinear term f x t satisfies some appropriate conditions related to a certain kato class of functions k m n r n

positive solution of singular dirichlet boundary value problems for second order differential equation system this paper investigates the existence of positive solutions of singular dirichlet boundary value problems for second order differential system a necessary and sufficient condition for the existence of c 0 1 c 0 1 positive solutions as well as c1 0 1 c1 0 1 positive solutions is given by means of the method of lower and upper solutions and the fixed point theorems our nonlinearity fi t x1 x2 may be singular at x1 0 x2 0 t 0 and or t 1 i 1 2

an existence and uniqueness result for the singular lane emden fowler equation abstract using a fixed point theorem of general concave operators we present in this paper criteria which guarantee the existence and uniqueness of positive solutions for the singular lane emden fowler equation

on a new kato class and positive solutions of dirichlet problems for the fractional laplacian in bounded domains abstract the purpose of this paper is to extend some results of the potential theory of an elliptic operator to the fractional laplacian 2 0 2 in a bounded c 1 1 domain d in r n in particular we introduce a new kato class k d and we exploit the properties of this class to study the existence of positive solutions of some dirichlet problems for the fractional laplacian

asymptotic behavior of positive solutions of a nonlinear dirichlet problem abstract we take up the existence and the asymptotic behavior of a classical solution to the following semilinear dirichlet problem u a x g u x u 0 in u 0 where is a c 1 1 bounded domain in r n n 2 and the function a belongs to c l o c 0 1 such that there exist c 1 c 2 0 satisfying for each x c 1 x 1 exp x z 1 s s d s a x c 2 x 2 exp x z 2 s s d s where d i a m x d i s t x 1 2 2 and for i 1 2 z i is a continuous function on 0 with z i 0 0 our arguments are based on the sub supersolution method with karamata regular variation theory

green function for schrodinger operator and conditioned feynman kac gauge consider the schrodinger operator h 2 q in a bounded c1 1 domain d with q klocd d q satisfies the condition sup spec 2 q d v x y g x y e y x exp 0 t q x t d t x y d where xt is the conditioned brownian motion starting at x and ending at y with the life time t and v g can be extended to a continuous and positive function on d d specifically we give the comparison theorem there exists a constant c c d q 0 such that 1 c g x y v x y c g x y x y d another result is that v nz x z x d z d exists and represents the poisson kernel for the dirichlet problem corresponding to schrodinger operator h

existence and optimal estimates of solutions for singular nonlinear dirichlet problems abstract by applying the properties of the unique classical solution to the singular boundary value problem on half line p s g p s p s 0 s 0 p 0 0 lims p s b 0 and constructing the new comparison functions they show the existence and the optimal global estimates of solutions to singular nonlinear dirichlet problems u k x g u u 0 x u 0 where is a bounded domain with smooth boundary in rn g s is nonincreasing and positive in 0 1 g t dt and lim s 0 g s k c is positive in and may be singular or zero on the boundary

existence and estimates of solutions for singular nonlinear elliptic problems for both bounded and unbounded domain in r d 3 with smooth boundary the function 0 0 is measurable and satisfies some appropriate hypotheses detailed below the problem 1 has been studied extensively for the special nonlinearity x t p x t 0 indeed in 7 edelson proved the existence of a positive solution u c2 loc r decaying as x 2 n as x provided that p c loc r 0 0 for x 0 and satisfying the growth condition

new fixed point theorems for mixed monotone operators and local existence uniqueness of positive solutions for nonlinear boundary value problems abstract in this article we present a new fixed point theorem for a class of general mixed monotone operators which extends the existing corresponding results moreover we establish some pleasant properties of nonlinear eigenvalue problems for mixed monotone operators based on them the local existence uniqueness of positive solutions for nonlinear boundary value problems which include neumann boundary value problems three point boundary value problems and elliptic boundary value problems for lane emden fowler equations is proved the theorems for nonlinear boundary value problems obtained here are very general

classical and weak solutions of a singular semilinear elliptic problem the singular semilinear elliptic equation u p x f u 0 is shown to have a unique positive classical solution inrnthat decays to zero at ifp x is simply a nontrivial nonnegative continuous function satisfying 0 t max x t p x dt providedfis a non increasing continuously differentiable function on 0 it is also shown that the equation has a unique weakh10 solution on a bounded domain provided e0 f s ds andp x l2

exact asymptotic behavior near the boundary to the solution for singular nonlinear dirichlet problems abstract we take up the existence and the exact asymptotic behavior near the boundary of the unique classical solution to a singular dirichlet problem u a x g u x u 0 in u 0 where is a c 1 1 bounded domain in r n n 2 and the function a is singular near the boundary and belongs to the kato class k

on the subordinate killed b m in bounded domains and existence results for nonlinear fractional dirichlet problems we take up in this paper the existence of positive continuous solutions for some nonlinear boundary value problems with fractional differential equation based on the fractional laplacian delta d frac alpha 2 associated to the subordinate killed brownian motion process z alpha d in a bounded c 1 1 domain d our arguments are based on potential theory tools on z alpha d and properties of an appropriate kato class of functions k d

asymptotic behavior of positive solutions of a semilinear polyharmonic problem in the unit ball abstract let m be a positive integer we investigate the existence and the asymptotic behavior of positive continuous solutions to the following semilinear polyharmonic boundary value problem in the unit ball b of r n n 2 m u a x u lim x 1 u x 1 x m 1 0 where 1 1 and a is a positive measurable function in b such that there exists c 0 satisfying for each x b 1 c a x 1 x exp 1 x z s s d s c 1 m 1 1 and z is a continuous function on 0 with z 0 0

the asymptotic behaviour of the unique solution for the singular lane emden fowler equation abstract by karamata regular variation theory and constructing comparison functions we show the exact asymptotic behaviour of the unique classical solution u c 2 c near the boundary to a singular dirichlet problem u k x g u u 0 x u 0 where is a bounded domain with smooth boundary in r n g c 1 0 0 lim t 0 g t g t for each 0 for some 0 and k c loc for some 0 1 is nonnegative on which is also singular near the boundary

the online median problem we introduce a natural variant of the metric uncapacitated k median problem that we call the online median problem whereas the k median problem involves optimizing the simultaneous placement of k facilities the online median problem imposes the following additional constraints the facilities are placed one at a time a facility cannot be moved once it is placed and the total number of facilities to be placed k is not known in advance the objective of an online median algorithm is to minimize the competitive ratio that is the worst case ratio of the cost of an online placement to that of an optimal offline placement our main result is a constant competitive algorithm for the online median problem running in time that is linear in the input size in addition we present a related though substantially simpler constant factor approximation algorithm for the metric uncapacitated facility location problem that runs in time linear in the input size the latter algorithm is similar in spirit to the recent primal dual based facility location algorithm of jain and vazirani but our approach is more elementary and yields an improved running time while our primary focus is on problems which ask us to minimize the weighted average service distance to facilities we also show that our results can be generalized to hold to within constant factors for more general objective functions for example we show that all of our approximation results hold to within constant factors for the k means objective function

two stage quadratic games under uncertainty and their solution by progressive hedging algorithms a model of a two stage n person noncooperative game under uncertainty is studied in which at the first stage each player solves a quadratic program parameterized by other players decisions and