on an inequality by n trudinger and j moser and related elliptic equations it has been shown by trudinger and moser that for normalized functions u of the sobolev space w 1 n whereis a bounded domain in r n one has r exp njuj n n 1 dx cn where n is an explicit constant depending only on n and cn is a constant depending only on n and carleson and chang proved that there exists a corresponding extremal function in the case that is the unit ball in r n in this paper we give a new proof a generalization and a new interpretation of this result in particular we give an explicit sequence that is maximizing for the above integral among all normalized concentrating sequences as an application the existence of a nontrivial solution for a related elliptic equation with trudinger moser growth is proved c 2002 john wiley sons inc

elliptic problems with nonlinearities indefinite in sign abstract let r n n 3 be a bounded open set with smooth boundary and r we study the dirichlet problem formula with 1 q p and h k l nonnegative functions we prove existence nonexistence and multiplicity results depending on and according to the integrability properties of the ratio k p 1 h q 1

a mixture model for occupational exposure mean testing with a limit of detection sul lhiary information from detectable exposure measurements randomly sampled from a left truncated log normal distribution may be used to evaluate the distribution of nondetectable values that fall below all analytic limit of detection if the proportion of nondetects is larger than expected under log normality alternative models to account for these unobserved data should be considered we discuss one such model that incorporates a mixture of true zero exposures and a log normal distribution with possible left censoring previously considered ill a different context by manitou and halsey 1995 biometrics 51 1570 1578 a particular relationship is demonstrated between maximum likelihood parameter estimates based on this mixture model and those assuming either left tnlllcated or left censored data these results emphasize the heed for caution when choosing a model to fit data involving nondetectable values a one sided likelihood ratio test for colli paring illeall exposure under the mixture model to an occupational exposure limit is then developed and evaluated via simulations all example demonstrates the potential impact of specifying an incorrect model for the ilondetectable value

positive and negative solutions for a class of kirchhoff type problems on unbounded domain abstract in this paper we study the existence of multiple positive and negative solutions for a class of kirchhoff type problems on an unbounded domain using the variational methods we show that our problem admits at least two positive and two negative solutions under different conditions a cut off functional is utilized to obtain the bounded palais smale sequences

nontrivial solutions of kirchhoff type problems via the yang index abstract we obtain nontrivial solutions of a class of nonlocal quasilinear elliptic boundary value problems using the yang index and critical groups

adams type inequalities and related elliptic partial differential equations in dimension four abstract motivated by ruf sani s recent work we prove an adams type inequality and a singular adams type inequality in the whole four dimensional euclidean space as applications of those inequalities a class of elliptic partial differential equations are considered existence of nontrivial weak solutions and multiplicity results are obtained via the mountain pass theorem and the ekeland s variational principle this is a continuation of our previous work about singular trudinger moser type inequality

existence of a positive solution to kirchhoff type problems without compactness conditions abstract the existence of a positive solution to a kirchhoff type problem on r n is proved by using variational methods and the new result does not require usual compactness conditions a cut off functional is utilized to obtain the bounded palais smale sequences

existence and bifurcation behavior of positive solutions for a class of kirchhoff type problems abstract we study the existence of positive solutions for the nonlinear kirchhoff type equation a b u 2 d x u u u 3 with dirichlet boundary condition where r are two real parameters we provide a description of a two dimensional set in the plane which correspond to the existence of positive solutions for the above kirchhoff type equation

multiple positive solutions to a class of kirchhoff equation on r3 with indefinite nonlinearity abstract we study the existence and multiplicity of positive solutions to a class of kirchhoff equation 1 b r 3 u x 2 d x u x u x v x u x p 2 u x k x u x u x u h 1 r 3 when v x is sign changing in r 3 we prove that there is 0 such that the kirchhoff equation possesses at least one positive solution for 0 1 and at least two positive solutions for 1 1 here 1 is the first eigenvalue of u u k x u u h 1 r 3 a positive eigenfunction corresponding to 1 is denoted by e 1 an interesting phenomena is that the term b r 3 u 2 d x u can be used to compete with the indefinite nonlinearity and we do not assume the condition r 3 v x e 1 p d x 0 which has been shown to be a necessary condition to the existence of positive solutions for semilinear elliptic equations with indefinite nonlinearity alama and tarantello 1993

estimation of distributional parameters for censored trace level water quality data 1 estimation techniques a recurring difficulty encountered in investigations of many metals and organic contaminants in ambient waters is that a substantial portion of water sample concentrations are below limits of detection established by analytical laboratories several methods were evaluated for estimating distributional parameters for such censored data sets using only uncensored observations their reliabilities were evaluated by a monte carlo experiment in which small samples were generated from a wide range of parent distributions and censored at varying levels eight methods were used to estimate the mean standard deviation median and interquartile range criteria were developed based on the distribution of uncensored observations for determining the best performing parameter estimation method for any particular data det the most robust method for minimizing error in censored sample estimates of the four distributional parameters over all simulation conditions was the log probability regression method with this method censored observations are assumed to follow the zero to censoring level portion of a lognormal distribution obtained by a least squares regression between logarithms of uncensored concentration observations and their z scores when method performance was separately evaluated for each distributional parameter over all simulation conditions the log probability regression method still had the smallest errors for the mean and standard deviation more but the lognormal maximum likelihood method had the smallest errors for the median and interquartile range when data sets were classified prior to parameter estimation into groups reflecting their probable parent distributions the ranking of estimation methods was similar but the accuracy of error estimates was markedly improved over those without classification less

existence of solutions for a higher order kirchhoff type problem with exponential critical growth abstract a higher order kirchhoff type equation r 2 m m u 2 0 m 1 a x u 2 d x m u 0 m 1 1 a x u f x u x h x in r 2 m is considered in this paper we assume that the nonlinearity of the equation has exponential critical growth and prove that for a positive which is small enough there are two distinct nontrivial solutions to the equation when 0 we also prove that the equation has a nontrivial mountain pass type solution

classical solutions for some higher order semilinear elliptic equations under weak growth conditions we want to explain the crucial difference between the second order m 1 and higher order m 1 case if m 1 a very satisfactory result is known indeed the sign condition 2 alone is sufficient to ensure classical solvability of the dirichlet problem 1 there are two very strong devices in the theory of second order elliptic equations which make the proof of the needed a priori maximum estimate an easy exercise

positive solutions for a quasilinear elliptic equation of kirchhoff type this paper is concerned with the existence of positive solutions to the class of nonlocal boundary value problems of the type m w u 2dx du f x u email protected u 0 email protected email protected where w is a smooth bounded domain of n m is a positive function and f has subcritical growth

existence of solutions to nonlinear subcritical higher order elliptic dirichlet problems abstract we consider the 2 m th order elliptic boundary value problem l u f x u on a bounded smooth domain r n with dirichlet boundary conditions on the operator l is a uniformly elliptic linear operator of order 2 m whose principle part is of the form i j 1 n a i j x 2 x i x j m we assume that f is superlinear at the origin and satisfies lim s f x s s q h x lim s f x s s q k x where h k c are positive functions and q 1 is subcritical by combining degree theory with new and recently established a priori estimates we prove the existence of a nontrivial solution

infinitely many positive solutions for kirchhoff type problems abstract this paper is concerned with the existence of infinitely many positive solutions to a class of kirchhoff type problem a b u 2 d x u f x u in and u 0 on where is a smooth bounded domain of r n a b 0 0 and f r r is a caratheodory function satisfying some further conditions we obtain a sequence of a e positive weak solutions to the above problem tending to zero in l with f being more general than that of k perera z zhang nontrivial solutions of kirchhoff type problems via the yang index j differential equations 221 2006 246 255 z zhang k perera sign changing solutions of kirchhoff type problems via invariant sets of descent flow j math anal appl 317 2006 456 463

the nehari manifold for a kirchhoff type problem involving sign changing weight functions abstract this paper examines a class of kirchhoff type equations that involve sign changing weight functions using nehari manifold and fibering map the existence of multiple positive solutions is established

multiplicity of positive solutions for some quasilinear elliptic equation in rnwith critical sobolev exponent consider the equation pu g x u p 2u f x u p 2u 1 in rn where 1 0 be the principal eigenvalue of pu g x u p 2uinrn rng x u p 0 2 withu 1 0 the associated eigenfunction we prove i equation 1 has at least one positive solution if 0 1 ii suppose rn f x u 1 p 1such that 1 has at least one positive solution for 1 0 moreover ifp 2 there exists 0 1such that 1 has at least two positive solutions for 1 0

multiple solutions for kirchhoff type equations in rn this paper is devoted to the existence of infinitely many solutions for a class of kirchhoff type equations setting on rn based on the minimax methods in critical point theory we obtain infinitely many large energy and small energy solutions which unify and sharply improve the recent results of wu existence of nontrivial solutions and high energy solutions for schrodinger kirchhoff type equations in rn nonlinear anal real world appl 12 1278 1287 2011

a uniqueness result for a nonlocal equation of kirchhoff type and some related open problem abstract we study the uniqueness of solution for the following boundary value problem involving a nonlocal equation of kirchhoff type a b u 2 d x u f u in u 0 here is a bounded open set in r n with smooth boundary a b are positive real numbers and f r r is a continuous function in particular we give an answer to an open problem recently proposed by b ricceri

existence and concentration result for the kirchhoff type equations with general nonlinearities in this paper we study the existence and concentration behaviors of positive solutions to the kirchhoff type equations varepsilon 2 m left varepsilon 2 n int mathbf r n nabla u 2 mathrm d x right delta u v x u f u quad rm in mathbf r n quad u in h 1 mathbf r n n geqq 1 where m and v are continuous functions under suitable conditions on m and general conditions on f we construct a family of positive solutions u varepsilon varepsilon in 0 tilde varepsilon which concentrates at a local minimum of v after extracting a subsequence ek

existence and concentration of positive solutions for kirchhoff type problems with a steep well potential abstract the existence and concentration of positive solutions for a class of kirchhoff type problems with a nonnegative steep well potential in r 3 are obtained via variational methods

numerical analysis and computing of a non arbitrage liquidity model with observable parameters for derivatives abstract this paper deals with the numerical analysis and computing of a nonlinear model of option pricing appearing in illiquid markets with observable parameters for derivatives a consistent monotone finite difference scheme is proposed and a stability condition on the stepsize discretizations is given

existence and convergence of solutions for nonlinear biharmonic equations on graphs abstract in this paper we first prove some propositions of sobolev spaces defined on a locally finite graph g v e which are fundamental when dealing with equations on graphs under the variational framework then we consider a nonlinear biharmonic equation 2 u u a 1 u u p 2 u on g v e under some suitable assumptions we prove that for any 1 and p 2 the equation admits a ground state solution u moreover we prove that as the solutions u converge to a solution of the equation 2 u u u u p 2 u in u 0 on where x v a x 0 is the potential well and denotes the boundary of

existence results of positive solutions of kirchhoff type problems abstract in the present paper we use variational methods to prove two existence results of positive solutions of the following kirchhoff type problems a b u 2 u f x u in u 0 on one deals with the asymptotic behaviors of f near zero and infinity and the other deals with 4 superlinear of f at infinity

existence of positive solutions to quasi linear elliptic equations with exponential growth in the whole euclidean space abstract in this paper a quasi linear elliptic equation in the whole euclidean space is considered the nonlinearity of the equation is assumed to have exponential growth or have critical growth in view of trudinger moser type inequality under some assumptions on the potential and the nonlinearity it is proved that there is a nontrivial positive weak solution to this equation also it is shown that there are two distinct positive weak solutions to a perturbation of the equation the method of proving these results is combining trudinger moser type inequality mountain pass theorem and ekeland s variational principle

isolated singularities of polyharmonic inequalities we study nonnegative classical solutions u of the polyharmonic inequality mu 0inb1 0 0 rn we give necessary and sufficient conditions on integers n 2 and m 1 such that these solutions u satisfy a pointwise a priori bound as x 0 in this case we show that the optimal bound for u is u x o x asx 0 where is the fundamental solution of in rn

l moment diagrams for censored observations observed data sets containing values above or below the analytical threshold of measuring equipment are referred to as censored such data are frequently encountered in quality and quantity monitoring applications of water soil and air samples most of the previous literature on the statistical analysis of censored data relates to the problems of moment parameter and quantile estimation methods such estimation methods usually assume an underlying probability distribution few goodness of fit methods exist for censored data we introduce l moment diagrams for the evaluation of the goodness of fit of alternative distributional hypotheses for left censored data experiments with artificial censored data sets document the conditions under which l moment diagrams should be useful our approach like hosking s 1995 approach for right censoring derived l moment diagrams for left censored observations from partial probability weighted moments

singular perturbed kirchhoff type problem with critical exponent abstract in the present paper we consider the following kirchhoff type problem with critical exponent e 2 a e b r 3 u 2 d x u v x u f u in r 3 u h 1 r 3 by variational methods we show the existence of the positive solutions concentrating around global minima of the potential v x as e 0 we do not need the monotonicity of the function f 3

positive solution for a superlinear kirchhoff type problem with a parameter abstract this paper is devoted to a nonlinear kirchhoff type problem depending on a real function and a nonnegative parameter on a smooth bounded domain of r n n 2 we show that if the nonlinearity is subcritical and superlinear at zero and infinity then there exists a positive value such that the kirchhoff type problem has at least one positive solution for any value of the parameter less than the positive value the main tools are variational methods critical point theory and iterative techniques

sign changing solutions of kirchhoff type problems via invariant sets of descent flow we obtain sign changing solutions of a class of nonlocal quasilinear elliptic boundary value problems using variational methods and invariant sets of descent flow

sign changing and multiple solutions of kirchhoff type problems without the p s condition abstract the existence of nontrivial solutions of kirchhoff type equations is an important nonlocal quasilinear problem in this paper we use minimax methods and invariant sets of descent flow to prove two interesting existence theorems for the following 4 superlinear kirchhoff type problems without the p s condition one concerning the existence of a nontrivial solution and the other one concerning the existence of sign changing solutions and multiple solutions a b u 2 u f x u in u 0 on

multiplicity of solutions for p x polyharmonic elliptic kirchhoff equations abstract in this paper we establish the existence of an unbounded sequence of solutions for a class of quasilinear elliptic p x polyharmonic kirchhoff equations including the new delicate degenerate case not yet covered in the literature the main tool is the symmetric mountain pass theorem of ambrosetti and rabinowitz

exponential problem on a compact riemannian manifold without boundary abstract let m g be an n dimensional compact riemannian manifold without boundary a trudinger moser type inequality says that sup u w 1 n 1 m e n u n n 1 d v g where u w 1 n is the usual sobolev norm of u w 1 n m n n n 1 1 n 1 and n 1 is the area of the unit sphere s n 1 using this inequality when e 0 is small enough we establish sufficient conditions under which the quasilinear equation n u u n 2 u f x u e h x has at least two nontrivial weak solutions in w 1 n m where n u div g u n 2 u f x u behaves like e u n n 1 as u for some 0 and h 0 belongs to the dual space of w 1 n m

existence and multiplicity of solutions to equations of n laplacian type with critical exponential growth in rn in this paper we deal with the existence of solutions to the nonuniformly elliptic equation of the form 0 1 div a x u v x u n 2u f x u x eh x in rn when f rn r r behaves like exp u n n 1 when u and satisfies the ambrosetti rabinowitz condition in particular in the case of n laplacian i e a x u u n 2 u we obtain multiplicity of weak solutions of 0 1 moreover we can get the nontriviality of the solution in this case when e 0 finally we show that the main results remain true if one replaces the ambrosetti rabinowitz condition on the nonlinearity by weaker assumptions and thus we establish the existence and multiplicity results for a wider class of nonlinearity see section 7 for more details

multiple solutions for n kirchhoff type problems with critical exponential growth in rn abstract in this paper we consider a class of nonlocal n kirchhoff type problems involving a nonlinearity term having critical exponential growth by applying variational methods together with trudinger moser inequality in whole r n we obtain the existence of at least two positive solutions

stochastic models applied to cluster analysis of censored water quality data a clustering method is presented which is appropriate when a mixture of multivariate distributions is used to model the data the procedure assumes that the measurements made on several independent characteristics follow probability distributions of different nature e g normal poisson multinomial likelihood methods are used to both estimate the parameters of the models and perform the cluster analysis the procedure is applicable when the data are subject to censoring a data set on the concentrations of volatile halocarbon contaminant in lake ontario is used to illustrate the methods of this paper

existence and concentration of sign changing solutions to kirchhoff type system with hartree type nonlinearity abstract in this paper we discuss the existence and the concentration of sign changing solutions to a class of kirchhoff type systems with hartree type nonlinearity in r 3 by the minimization argument on the sign changing nehari manifold and a quantitative deformation lemma we prove that the system has a sign changing solution moreover concentration behaviors of sign changing solutions are obtained when the coefficient of the potential function tends to infinity specially our results cover general schrodinger equations kirchhoff equations and schrodinger poisson systems

min max level estimate for a singular quasilinear polyharmonic equation in r2m abstract using the framework first presented by ruf and sani we give a proof of an adams type inequality which can be applied to the functional j u 1 2 r 2 m m u 2 0 m 1 a x u 2 d x r 2 m f x u x d x r 2 m h u d x under two kinds of assumptions on the nonlinearity we estimate the min max level of the functional as an application a multiplicity result for the related singular quasilinear elliptic equation is proved

numerical analysis and simulation of option pricing problems modeling illiquid markets this paper deals with the numerical analysis and simulation of nonlinear black scholes equations modeling illiquid markets where the implementation of a dynamic hedging strategy affects the price process of the underlying asset a monotone difference scheme ensuring nonnegative numerical solutions and avoiding unsuitable oscillations is proposed stability properties and consistency of the scheme are studied and numerical simulations involving changes in the market liquidity parameter are included

the critical problem of kirchhoff type elliptic equations in dimension four abstract we study the following kirchhoff type elliptic problem p a b u 2 d x u u q u 3 u 0 in u 0 on where r 4 is a bounded domain with smooth boundary moreover we assume a 0 b 0 and 1 q 3 in this paper we prove the existence of solutions of p our tools are the variational method and the concentration compactness argument for ps sequences

existence and multiplicity of solutions for a superlinear kirchhoff type equations with critical sobolev exponent in r n abstract in this article we consider a class of superlinear kirchhoff type equations with critical growth a b r n u 2 d x u u 2 1 k x u q 1 x r n u d 1 2 r n where 0 n 4 2 q 2 2 2 n n 2 a b 0 and a b 0 k satisfies some conditions by using the variational method the multiplicity of positive solutions and infinitely many pairs of solutions are obtained

inferences about the mean from censored water quality data several methods which are commonly used for making inferences about the levels of many metals and organic contaminants in ambient waters from type i censored data are critically evaluated and their applications are illustrated using the concentrations of polychlorinated biphenyls in water samples from the niagara river difficulties encountered in the application of all the methods except the method of maximum likelihood include ignoring the information available about the analytical detection limit and the unavailability of the standard errors for these estimates under the assumption that the distribution of the data is lognormal the log regression method is modified to produce estimates of the censored values which use the detection limit information and the properties of the maximum likelihood estimates for the lognormal mean are derived furthermore an approximate confidence interval for the lognormal mean is given and its use for estimating the load to and from the niagara river is illustrated the above methods along with a modification to the method of maximum likelihood are evaluated using monte carlo simulation