uniqueness of positive radial solutions for infinite semipositone p laplacian problems in exterior domains abstract we prove uniqueness and asymptotic behavior of positive radial solutions to the p laplacian problem p u k x f u in x r 0 u 0 on x r 0 u x 0 as x where x r n x r 0 0 n p f 0 r is continuous f u u q at for some q 0 p 1 with possible infinite semipositone structure at 0 and is a large parameter

monotone approximations of unstable solutions assume that for a nonlinear two point problem we have a subsolution lying above a supersolution it is well known that in general one cannot conclude the existence of a solution in between however if one restricts the growth of the nonlinearity then under some restrictions on the super and subsolution one gets both existence of a solution and two sequences of monotone approximations in addition we can assert some qualitative property of the solution

semipositone problems with falling zeros on exterior domains abstract we study boundary value problems of the form u k x f u x u 0 if x r 0 u 0 as x where is a positive parameter u div u is the laplacian of u x r n n 2 x r 0 k belongs to a class of c 1 functions such that lim r k r 0 and f belongs to a class of c 1 functions which are negative at the origin and have falling zeros we discuss the existence and uniqueness of nonnegative radial solutions when is large

on singular sturm liouville boundary value problems we consider the existence of positive solutions for the boundary value problem q t u f t u 0 r t r au r b 1 q r u r 0 cu r d 1 q r u r 0 where u u p 2 u p 1 0 f is p superlinear or p sublinear at and is allowed to become at u 0 our results unify and extend many known results in the literature

uniqueness of positive radial solutions for a class of semipositone problems on the exterior of a ball abstract we study positive radial solutions to u k x f u x e where 0 is a parameter e x r n x r 0 r 0 0 n 2 is the laplacian operator k c r 0 0 satisfies k r 1 r n 0 for r 1 and f c 1 0 r is a concave increasing function satisfying lim s f s s 0 and f 0 0 semipositone we are interested in solutions u such that u 0 as x and satisfy the nonlinear boundary condition u c u u 0 if x r 0 where is the outward normal derivative and c c 0 0 is an increasing function we will establish the uniqueness of positive radial solutions for large values of the parameter

uniqueness of nonnegative solutions for semipositone problems on exterior domains abstract we consider the problem u k x f u x u 0 if x r 0 u 0 as x where is a positive parameter u div u is the laplacian of u x r n n 2 x r 0 k c 1 r 0 0 is such that lim r k r 0 and f c 1 0 r is a concave function which is sublinear at and f 0 0 we establish the uniqueness of nonnegative radial solutions when is large

uniqueness theorems for positive radial solutions of quasilinear elliptic equations in a ball we establish a new pohozaev type identity and use it to prove a theorem on the uniqueness of positive radial solutions to the quasilinear elliptic problem div u m 2emsp14 u f u 0 inb andu 0 on b wherebis a finite ball in rn n 3 and 1 0 and 1 p q n 2 n 2 has a unique positive solution whenn 6 this gives a complete answer to an open problem raised by brezis and nirenberg in 1983 in the casen 6 we shall also derive some partial results to the open problem in the casesn 3 4 5

exact multiplicity for degenerate two point boundary value problems with p convex nonlinearity the exact number of positive solutions of a degenerate quasilinear two point boundary value problem is investigated for the generalization of earlier results concerning the nondegenerate case with convex nonlinearity suitably defined p convex nonlinearities are considered strictly p convex c2 functions having a non negative root are classified according to the shape of the bifurcation diagram of positive solutions versus the length of the interval we have uniqueness when f 0 0 and 0 1 or 2 solutions when f 0 0 similarly to the non degenerate case provided that the number of solutions is finite however now there may also occur a continuum of solutions connected to a dead core type phenomenon the proof of our results relies on the shooting method for the characterization of the shape of the time map in contrast to the non degenerate case the shooting method does not determine directly the number of solutions owing to the lack of uniqueness of the corresponding ivp exact conditions on the uniqueness of the ivp and in the case of non uniqueness the number and types of its local solutions are given based on this all the positive solutions of the bvp can be compiled

uniqueness of positive solutions for a class of semilinear elliptic systems we prove existence and uniqueness of positive solutions for the system u f v v g u in b u v 0 on b where b is the open unit ball in rn f g r r and f x xp g x xq at for some p q 0 with pq 1

positive radial solutions to classes of nonlinear elliptic systems on the exterior of a ball abstract we study positive radial solutions to n n nonlinear elliptic systems of the form u i k i x f i u i 1 in x e u i 0 on x r 0 u i 0 as x where i 1 2 n u 1 u n 1 u div u 0 n 2 r 0 0 e x r n x r 0 k i c r 0 0 and f i c 0 r we establish existence multiplicity and uniqueness results for various nonlinearities f i

boundary behavior of solutions of monge ampere equations with singular righthand sides abstract in this paper by karamata regular variation theory and constructing super and subsolutions we obtain the asymptotic boundary behavior of convex solutions of monge ampere equations with singular righthand sides

harnack inequality for nonlinear elliptic equations with strong absorption abstract we develop harnack inequalities for two different types of equations first we consider a fully nonlinear uniformly elliptic equation related to the pucci s maximal and the minimal operators next we consider a quasilinear equation related to the p laplacian in both cases we consider lower order terms of keller osserman type although the equations considered are quite different we employ a unified method to approach both problems and the results we find are similar

entire solutions of fully nonlinear elliptic equations with a superlinear gradient term in this paper we consider second order fully nonlinear operators with an additive superlinear gradient term like in the pioneering paper of brezis for the semilinear case we obtain the existence of entire viscosity solutions defined in all the space without assuming global bounds a uniqueness result is also obtained for special gradient terms subject to a convexity concavity type assumption where superlinearity is essential and has to be handled in a different way from the linear case

evolution of positive solution curves in semipositone problems with concave nonlinearities abstract we study the existence multiplicity and stability of positive solutions to u x f u x for x 1 1 u 1 0 u 1 where 0 and f 0 r is monotonically increasing and concave with f 0 0 we obtain the exact number of positive solutions as a function of f t t we follow several families of nonlinearities f for which f limt f t 0 and study how the positive solution curves to the above problem evolve also we give examples where our results apply this work extends the work of a castro and r shivaji 1988 proc roy soc edinburgh sect a108 291 302 and s h wang 1994 proc roy soc edinburgh124 no 3 507 515 by obtaining sharper results and also gives a complete study of positive solutions for concave semipositone nonlinearities

uniqueness theorems for negative radial solutions of k hessian equations in a ball abstract in this paper we apply a pohozaev type identity and the monotone separation techniques to prove the uniqueness of negative radial solutions of k hessian equations in a finite ball

uniqueness of positive solutions for a class of semipositone elliptic systems abstract we consider the system u f v x v g u x u 0 v x where is a ball in r n n 1 and is its boundary are positive parameters and f g are smooth functions that are negative at the origin and f x x p and g x x q for large x for some p q 0 with p q 1 we establish the uniqueness of positive solutions when the parameters and are large

structure of positive radial solutions of semilinear elliptic equations abstract we study the positive radial solutions of a semilinear elliptic equation u f u 0 wheref u has a supercritical growth order for smallu 0 and a subcritical growth order for largeu by showing the uniqueness of positive solutions behaving likeo x 2 n at infinity we give an almost complete description for the structure of positive radial solutions as a consequence we also prove the uniqueness of positive solutions of the nonlinear dirichlet problem for the equation in a finite ball

singular semilinear elliptic problems with asymptotically linear reaction terms abstract we consider the problem u k x f u in b 1 c u 0 on b 1 u x 0 as x where b 1 c x r n x 1 n 2 is a positive parameter k belongs to a class of functions which satisfy certain decay assumptions and f belongs to a class of functions which are asymptotically linear and may be singular at the origin we prove the existence of positive solutions to such problems for certain values of parameter existence results to similar problems in r n are also obtained our existence results are proved using the schauder fixed point theorem and the method of sub and super solutions

existence and uniqueness of nontrivial radial solutions for k hessian equations abstract we prove existence and uniqueness of nontrivial radial solutions to the k hessian problem s k d 2 u f u in u 0 on where s k d 2 u is the k hessian operator of u is the open unit ball in r n f is a continuous function and may have k sublinear growth at with possible k superlinear growth at 0 and is a large parameter

boundary behavior of large solutions to the monge ampere equations with weights abstract this paper is concerned with the monge ampere equation det d 2 u x b x f u x x where is a strictly convex bounded smooth domain in r n with n 2 and b c which is positive in but may be vanishing on the boundary we find a new structure condition on f which plays a crucial role in the boundary behavior of strictly convex large solutions our results are obtained in a more general setting than those in cirstea and trombetti 2008 12 where f is regularly varying at infinity with index p n

weighted eigenvalue problems for hessian equations abstract in this paper we consider weighted eigenvalue problems for fully nonlinear elliptic equations involving hessian operators in particular we consider a singular weight which behaves like a hardy potential and we prove the existence of weak eigenfunctions

existence and estimates of solutions to a singular dirichlet problem for the monge ampere equation given a strictly convex smooth and bounded domain in rn we establish the existence of a negative convex solution in c c with zero boundary value to the singular monge ampere equation det d2u p x g u an associated dirichlet problem will be employed to provide a necessary and sufficient condition for the solvability of the singular boundary value problem estimates of solutions will also be given and regularity of solutions will be deduced from the estimates

some general effects of strong high frequency excitation stiffening biasing and smoothening abstract mechanical high frequency hf excitation provides a working principle behind many industrial and natural applications and phenomena this paper concerns three particular effects of hf excitation that may change the apparent characteristics of mechanical systems 1 stiffening by which the apparent linear stiffness associated with an equilibrium is changed along with derived quantities such as stability and natural frequencies 2 biasing by which the system is biased towards a particular state static or dynamic which does not exist or is unstable in the absence of the hf excitation and 3 smoothening referring to a tendency for discontinuities to be effectively smeared out by hf excitation illustrating first these effects for a few specific systems analytical results are provided that quantify them for a quite general class of mechanical systems this class covers systems that can be modelled by a finite number of second order ordinary differential equations generally non linear with periodically oscillating excitation terms of high frequency and small amplitude the results should be useful for understanding the effects in question in a broader perspective than is possible with specific systems for calculating effects for specific systems using well defined formulas and for possibly designing systems that display prescribed characteristics in the presence of hf excitation

on singular boundary value problems for the monge ampere operator abstract we consider different types of singular boundary value problems for the monge ampere operator the approach is based on existing regularity theory and a subsolution supersolution method nonexistence and uniqueness results are also given

a bernstein type theorem for parabolic k hessian equations abstract we are concerned with the characterization of entire solutions to the parabolic k hessian equation of the form u t f k d 2 u 1 in r n 0 we prove that for 1 k n any strictly convex monotone solution u u x t c 4 2 r n 0 to u t f k d 2 u 1 in r n 0 must be a linear function of t plus a quadratic polynomial of x under some growth assumptions on u

existence and non existence of solutions for a class of monge ampere equations abstract we study the boundary value problems for monge ampere equations det d 2 u e u in r n n 1 u 0 first we prove that any solution on the ball is radially symmetric by the argument of moving plane then we show there exists a critical radius such that if the radius of a ball is smaller than this critical value there exists a solution and vice versa using the comparison between domains we can prove that this phenomenon occurs for every domain finally we consider an equivalent problem with a parameter det d 2 u e t u in u 0 t 0 by using lyapunov schmidt reduction method we get the local structure of the solutions near a degenerate point by leray schauder degree theory a priori estimate and bifurcation theory we get the global structure

boundary behavior of k convex solutions for singular k hessian equations abstract we discuss the existence and boundary behavior of k convex solution to the singular k hessian problem s k d 2 u x b x f u x x u x 0 x where s k d 2 u k 1 2 n is the k hessian operator r n n 2 is a smooth bounded strictly convex domain here the weight function b x is not necessarily bounded on another interest is that f u as u 0 our approach mainly relies on karamata s regular variation theory and the construction of suitable sub and super solutions

a new method for computing the matrix exponential operation based on vector valued rational approximations in this paper a new method for computing the action of the matrix exponential on a vector e a tb where a is a complex matrix and t is a positive real number is proposed our approach is based on vector valued rational approximation where the approximants are determined by the denominator polynomials whose coefficients are obtained by solving an inexpensive linear least squares problem no matrix multiplications or divisions but matrix vector products are required in the whole process a technique of scaling and recurrence enables our method to be more effective when the problem is for fixed a b and many values of t we also give a backward error analysis in exact arithmetic for the truncation errors to derive our new algorithm preliminary numerical results illustrate that the new algorithm performs well

existence and multiplicity for negative solutions of k hessian equations abstract in this paper we give some existence results for k hessian equations by sub sup method and discuss the multiplicity of k hessian equations especially for the radial solutions in a ball

a uniqueness result for a semipositone p laplacian problem on the exterior of a ball abstract we consider steady state reaction diffusion equations on the exterior of a ball namely boundary value problems of the form p u k x f u in e u 0 on x r 0 u 0 when x where p z div z p 2 z 1 p n is a positive parameter r 0 0 and e x r n x r 0 here the weight function k c 1 r 0 satisfies k r 0 for r r 0 lim r k r 0 and the reaction term f c 0 c 1 0 is strictly increasing and satisfies f 0 0 semipositone lim sup s 0 s f s lim s f s lim s f s s p 1 0 and f s s q is nonincreasing on a for some a 0 and q 0 p 1 for a class of such steady state equations it turns out that every nonnegative radial solution is strictly positive in the exterior of a ball and exists for 1 we establish the uniqueness of this positive radial solution for 1

bifurcation and admissible solutions for the hessian equation abstract we study the following eigenvalue problem of k hessian equation s k d 2 u k f u in b u 0 on b global bifurcation result is established for this problem as applications of the bifurcation result we determine the intervals of for the existence nonexistence uniqueness and multiplicity of radially symmetric k admissible solutions for this problem

diffusive logistic equation with constant yield harvesting i steady states we consider a reaction diffusion equation which models the constant yield harvesting to a spatially heterogeneous population which satisfies a logistic growth we prove the existence uniqueness and stability of the maximal steady state solutions under certain conditions and we also classify all steady state solutions under more restricted conditions exact global bifurcation diagrams are obtained in the latter case our method is a combination of comparison arguments and bifurcation theory

on the stability properties of nonnegative solutions of semilinear problems with convex or concave nonlinearity 497 abstract we investigate the stability of nontrivial nonnegative stationary solutions of semilinear initial boundary value problems with convex or concave nonlinearity in the convex case f 0 0 implies instability in the concave case f 0 0 implies stability we also discuss the necessity of the sign condition on f 0

exact multiplicity of positive solutions of a semipositone problem with concave convex nonlinearity abstract we study the exact multiplicity of positive solutions and bifurcation diagrams of the dirichlet boundary value problem u x f u 0 1 x 1 u 1 u 1 0 where 0 is a bifurcation parameter f c 0 c 2 0 satisfies f 0 0 semipositone and f is concave convex on 0 and is asymptotic superlinear assuming additional suitable conditions on f on the u plane we give a classification of totally three qualitatively different bifurcation diagrams a reversed s shaped curve a broken reversed s shaped curve and a monotone decreasing curve our results improve those in j shi r shivaji exact multiplicity of solutions for classes of semipositone problems with concave convex nonlinearity discrete contin dyn syst 7 2002 559 571 we also give an application to determine completely the exact multiplicity of positive solutions and bifurcation diagrams of the problem with cubic nonlinearity u x u a u b u c 0 1 x 1 u 1 u 1 0 0 a b c

positive solution curves of semipositone problems with concave nonlinearities we consider the positive solutions to the semilinear equation where denotes a smooth bounded region in n n 1 and 0 here f 0 is assumed to be monotonically increasing concave and such that f 0 f lim t f t 0 we establish the stability and uniqueness of large positive solutions in terms of f t t when is a ball we determine the exact number of positive solutions for each 0 we also obtain the geometry of the branches of positive solutions completely and establish how they evolve this work extends and complements that of 3 7 where f 0

global smoothness for a singular monge ampere equation abstract the global c n 2 regularity for the graph of solutions to a singular monge ampere equation was established in 7 this equation arises in the study of hyperbolic affine spheres and is singular at the boundary a compatibility condition for higher regularity was found in 7 in this paper we verify the compatibility condition and obtain the global c regularity in the even dimensional case

existence and uniqueness for a class of quasilinear elliptic boundary value problems abstract we prove existence and uniqueness of positive solutions for the boundary value problem r n 1 u r n 1 f u u 0 u 1 0 where x x p 2x f x xp 1 may not be decreasing on 0 and is a large parameter

uniqueness of positive solutions for quasilinear elliptic equations when a parameter is large existence and uniqueness results are proved for positive solutions of a class of quasilinear elliptic equations in a domain n via a generalisation of serrin s sweeping principle in the case when is an annulus it is shown that the solution is radially symmetric

unbounded connected component of the positive solution set of nonlinear operator equation and its applications abstract in this paper by using global bifurcation theories we obtain some results for structure of positive solution set of some nonlinear equations with parameters as a result we obtain some existence results for positive solutions of the nonlinear operator equation the main result can be applied to various of differential boundary value problems to obtain the existence results for positive solutions without the assumption that the nonlinearities are positone

fully nonlinear elliptic equations for conformal deformations of chern ricci forms abstract we study a general class of fully nonlinear elliptic equations of second order on hermitian manifolds we derive a priori estimates and prove some existence results for these equations in particular we prove interior estimates for the complex hessian which in general do not hold for other types of equations we apply our results to draw geometric conclusions on conformal deformations of the mixed chern ricci forms a notion we introduced in the article our work is motivated by the close connections of these equations to problems in non kahler geometry and the fact that there have been increasing interests in fully nonlinear pde s beyond the complex monge ampere equation from complex geometry

uniqueness of positive solutions for a class of quasilinear problems abstract we prove uniqueness of positive solutions for the quasilinear problems p u f u in u 0 on where p u div u p 2 u p 1 is a bounded domain in r n with smooth boundary f is p sublinear at f u u p 1 is decreasing for large u and is a large positive parameter we also obtain the asymptotic behavior of the solution obtained as

branches of radial solutions for semipositone problems abstract we consider the radially symmetric solutions to the equation u x u x for x u x 0 for x where denotes the unit ball in r n n 1 centered at the origin and 0 here r r is assumed to be semipositone 0 0 there exists at most one positive solution u such that u belongs to the unbounded branch of positive solutions further when t tp k k 0 and 1

on the dirichlet problem for a class of augmented hessian equations abstract in this paper we consider the dirichlet problem for a new class of augmented hessian equations under sharp assumptions that the matrix function in the augmented hessian is regular and there exists a smooth subsolution we establish global second order derivative estimates for the solutions to the dirichlet problem in bounded domains the results extend the corresponding results in the previous paper 12 from the monge ampere type equations to the more general hessian type equations

large solutions to the monge ampere equations with nonlinear gradient terms existence and boundary behavior abstract in this paper we obtain conditions about the existence and boundary behavior of strictly convex solutions to the monge ampere equations with boundary blow up det d 2 u x b x f u x u q x u and det d 2 u x b x f u x 1 u q x u where is a strictly convex bounded smooth domain in r n with n 2 q 0 n or q 0 n b c which is positive in but may vanish or blow up on the boundary f c 0 f 0 0 and f is strictly increasing on 0 or f c r f s 0 s r and f is strictly increasing on r

an existence result on positive solutions for a class of semilinear elliptic systems consider the system where is a positive parameter and is a bounded domain in r n we prove the existence of a large positive solution for large when lim x f mg x x 0 for every m 0 in particular we do not need any monotonicity assumptions on f g nor any sign conditions on f 0 g 0

bivariate thiele type matrix valued rational interpolants a new method for the construction of bivariate matrix valued rational interpolants on a rectangular grid is introduced in this paper the rational interpolants are of the continued fraction form with scalar denominator in this respect the approach is essentially different from that of bose and basu 1980 where a rational matrix valued approximant with matrix valued numerator and denominator is used for the approximation of a bivariate matrix power series the matrix quotients are based on the generalized inverse for a matrix introduced by gu chuanqing and chen zhibing 1995 which is found to be effective in continued fraction interpolation a sufficient condition of existence is obtained some important conclusions such as characterisation and uniqueness are proven respectfully the inner connection between two type interpolating functions is investigated some examples are given so as to illustrate the results in the paper

w2 1 regularity for solutions of the monge ampere equation in this paper we prove that a strictly convex alexandrov solution u of the monge ampere equation with right hand side bounded away from zero and infinity is w 2 1 mathrm loc this is obtained by showing higher integrability a priori estimates for d 2 u namely d 2 u llog k l for any k

on the exterior dirichlet problem for hessian equations in this paper we establish a theorem on the existence of the solutions of the exterior dirichlet problem for hessian equations with prescribed asymptotic behavior at infinity this extends a result of caffarelli and li in 3 for the mongeampere equation to hessian equations

existence of solutions for semilinear problems with prescribed number of zeros on exterior domains abstract in this paper we prove the existence of an infinite number of radial solutions of u k r f u 0 on the exterior of the ball of radius r centered at the origin in r n such that lim r u r 0 with prescribed number of zeros where f r r is odd and there exists a 0 with f 0 on 0 f 0 on with f superlinear for large u and k r r with 0 n

vector valued thiele werner type osculatory rational interpolants a method of solving vector valued osculatory rational interpolation problems is presented by using the samelson inverse of vectors the rational interpolants are constructed in the form of thiele werner type continued fraction expression its interpolation structures are analyzed and important characteristic properties are obtained

synchronization of three non identical coupled exciters with the same rotating directions in a far resonant vibrating system abstract the aim of this paper is to investigate the synchronization of three non identical coupled exciters rotating with the same directions in a far resonant vibrating system of plane motion by deducing the dimensionless coupling equations of three exciters the synchronization problem is reduced to an existence and stability problems of finding the zero solutions for the averaged differential equations by introducing the dimensionless coupling torques and the characteristics of the corresponding limited functions the synchronization criterion for three exciters is obtained the stability criterion of synchronous states is derived it satisfies automatically the criterions of the generalized lyapunov and routh hurwitz the diversity feature of the vibrating system is discussed numerically both numerical and simulation results show that the structural parameters of the vibrating system can ensure the stability criterion of its synchronous operation

vector valued rational interpolants i it is shown explicitly that the samelson inverse may be used to define vector valued thiele type rational interpolants and the equivalence with claessens algorithm is established for the special case of vector valued pade approximants the general form of mccleod s theorem follows as a corollary

a new approach to the analysis of oscillations of one dimensional spatially periodic structures abstract a new approach to the analysis of oscillations of spatially periodic structures which is based on the method of direct separation of motions mdsm is proposed in the paper as an example of the approach application the study of oscillations of a string with variable cross section is conducted as the result analytical expressions for the eigenmodes and the eigenfrequencies of the system are obtained it is shown that modulation of the string cross section leads to a change of the eigenfrequencies as compared with their non modulated values and to the emergence of a spectrum of additional high eigenfrequencies which correspond to large wave lengths it is noted that application of the multiple scales method and other classical asymptotic methods for solving the considered equations is rather cumbersome and leads to erroneous results because an unknown parameter is present in them the fundamental difference between the mdsm and these methods is revealed in particular it is shown that the mdsm may be employed for solving equations without small parameter thereby the applicability range of this method turns out to be broader than the one of the classical asymptotic methods to illustrate the validity of this statement a classical problem about the stability of a pendulum with vibrating suspension axis is considered

existence of positive radial solutions for a semipositone elliptic equation abstract in this paper we study the existence of positive radial solutions for a semipositone elliptic equation u f u in u 0 on where is a ball or an annulus in r n with n 2 0 is a parameter and f is a continuous function satisfying the semipositone condition f 0 0 we give a weak and general sufficient condition on f for the existence of positive radial solutions for 0 large we show that the linearized operator at each solution has the nonnegative first eigenvalue these solutions are obtained as minimizers of a lagrangian functional

thiele type and lagrange type generalized inverse rational interpolation for rectangular complex matrices abstract a variety of matrix rational interpolation problems include the partial realization problem for matrix power series and the minimal rational interpolation problem for general matrix functions different from the previous work in this paper we consider a new method of matrix rational interpolation with rectangular real or complex interpolated matrices and distinct real or complex interpolation points based on an axiomatic definition for the generalized inverse matrix rational interpolants gmri gmri are constructed in the following two forms i thiele type continued fraction expression ii an explicit determinantal formula for the denominator scalar polynomials and for the numerator matrix polynomials which are of lagrange type expression as a direct application of gmri a matrix rational extrapolation is introduced

on the separation of fast and slow motions in mechanical systems with high frequency modulation of the dissipation coefficient abstract dynamics of a model mechanical system with fast and strong oscillations of the damping coefficient has been analyzed by fidlin 2005 6 he has performed the asymptotic analysis of the equation of motion of this system to conclude that these oscillations produce variation in its effective stiffness the present paper continues analysis of dynamics of the system in the regimes of motion treated by fidlin as well as in those left out in his asymptotic solution the results are compared with the results of the solution of the classical mathieu equation which features fast oscillations in the stiffness of a system the influence of stiffness and damping modulations on the stability of motion of corresponding oscillators is studied several engineering applications modeled by the system with oscillations of the damping coefficient are introduced analysis of motion of this system exemplifies how the method of direct separation of motions blekhman 2000 7 can be applied for solving equations with fast oscillating terms depending on the velocities some features of the application of the method of direct separation of motions in this case and in the similar ones are highlighted

the existence and asymptotic behavior of boundary blow up solutions to the k hessian equation abstract in this paper we consider the existence and asymptotic behavior of k convex solution to the boundary blow up k hessian problem s k d 2 u h x f u for x u x as dist x 0 where k 1 2 n s k d 2 u is the k hessian operator is a smooth bounded strictly convex domain in r n n 2 h c is positive in but is not necessarily bounded on and f is a smooth positive function that satisfies the so called keller osserman condition further results are obtained for the special case that is a ball our approach to show the existence and asymptotic behavior exploits the method of sub and super solutions and karamata regular variation theory

on a k hessian equation with a weakly superlinear nonlinearity and singular weights abstract consider the existence nonexistence and global estimates of k convex solutions to the boundary blow up k hessian problem s k d 2 u x h x u x k ln u x 0 for x u x as dist x 0 here k 1 2 n s k d 2 u is the k hessian operator 0 is a smooth bounded strictly convex domain in r n n 2 and h x is a positive weight function which is singular near the boundary we first give the existence and nonexistence results of k convex solution to the above boundary blow up problem on a larger range of h and then we show that there is a k convex solution provided that h x grows fast near and u k ln u grows slow at it turns out that this case is more difficult to handle than the case in which h x grows slow near and u k ln u grows fast at this needs some new ingredients in the arguments

an existence result on positive solutions for a class of p laplacian systems abstract consider the system p u f v in p v g u in u v 0 on where pz div z p 2 z p 1 is a positive parameter and is a bounded domain in rn with smooth boundary we prove the existence of a large positive solution for large when lim x f m g x 1 p 1 x p 1 0 for every m 0 in particular we do not assume any sign conditions on f 0 or g 0

critical dimension of a hessian equation involving critical exponent and a related asymptotic result abstract let s k 2 u 1 k n 2 be the k th hessian operator we study the problem formula where k is the critical exponent for s k and is a ball results generalizing those obtained by brezis nirenberg for the special case k 1 are established a discussion 8 on the asymptotic behavior of solutions of the problem as 0 is also included

positive solutions for infinite semipositone problems with falling zeros abstract we consider the positive solutions to the singular problem p u a u f u c u in u 0 on where 0 1 a 0 and c 0 are constants is a bounded domain with smooth boundary and f 0 r is a continuous function we assume that there exist m 0 a 0 p 1 such that a u m f u a u p for all u 0 a simple example of f satisfying these assumptions is f u u p for any p 1 we use the method of sub supersolutions to prove the existence of a positive solution of p when a 2 1 1 and c is small here 1 is the first eigenvalue of operator with dirichlet boundary conditions we also extend our result to classes of infinite semipositone systems