existence of positive entire solutions for a class of quasilinear elliptic equations d lt d r d d liix ax 1 6 i j n and i is a real constant by an entire solution of 1 1 we mean a function ii of class c rn which satisfies 1 1 in rn in recent years the existence of positive entire solutions of 1 1 has been studied under various structure hypotheses for l and f by many authors see e g 1 8 l 15 17 22 241 however as far as we are aware little is known about the existence of nonconstant entire solutions of eq 1 1 subject to the condition

ergodic actions of group extensions on von neumann algebras on considere des actions automorphes sur des algebres de von neumann d un groupe localement compact e donne comme une extension topologique o a e g o ou a est abelien compact et second denombrable on cherche a decrire et a classer des actions ergodiques de e quand g est fini

existence and asymptotic behavior of radially symmetric ground states of quasi linear singular elliptic equations existence and asymptotic behavior of entire positive solutions of a class of quasi linear elliptic equation div u p email protected u email protected x f u email protected r n is obtained under several hypotheses on the r x and f r we obtain the existence of positive entire solution asymptotic behavior is discussed by constructing an upper solution the results of this paper is new and extend previously known results

classification of minimal actions of a compact kac algebra with amenable dual on injective factors of type iii abstract we classify a certain class of minimal actions of a compact kac algebra with amenable dual on injective factors of type iii the structural analysis of type iii factors and the canonical extension of endomorphisms introduced by izumi are our main technical tools

compact ergodic groups of automorphisms it is shown that if g is a compact ergodic group of automorphisms on a unital c algebra a then the unique g invariant state is a trace hence if a is a von neumann algebra then it is finite and infective

the characteristic square of a factor and the cocycle conjugacy of discrete group actions on factors this paper has three main purposes rst to associate to any separable factor m a canonical commuting exact square of groups called the characteristic square ofm secondly to formulate at least in a preliminary way a methodology for approaching non smooth classi cation problems in analysis and thirdly to use this approach in conjunction with the characteristic square to complete the cocycle conjugacy classi cation of actions of discrete amenable groups on injective factors of type iii1 this result and previous works on the cocyle conjugacy classi cation problem combine to produce a cocycle conjugacy classi cation theorem which applies simultaneously to all countable amenable groups and all injective separable factors and in which the characteristic square and our new methodology play a crucial role the cocycle conjugacy classi cation problem has by now a long and distinguished history beginning with connes ground breaking outer conjugacy classi cation of single automorphisms of the approximately nite dimensional or afd factor r0 of type ii1 c3 in the intervening 20 years enormous progress has been made the rst step was taken by v jones j who identi ed the true nature of the cocycle conjugacy invariants and completed the classi cation of actions of nite groups on r0 ocneanu o re ned this work and introduced new analytic techniques enabling him to extend the classi cation to countable amenable groups acting on the semi nite afd factor r0 1 and to some actions on afd factors of type iii also sutherland and takesaki using groupoid and duality techniques originating in jt extended the classi cation results to invent math 132 331 380 1998

entire solutions of quasilinear elliptic equations abstract we consider the existence of entire solutions of a quasilinear elliptic equation div d u p 2 d u k x f u x r n where p 1 n n conditions of the existence of entire solutions have been obtained by different authors we prove an optimality of some of these results and give new sufficient conditions for the nonexistence of entire solutions

discrete time control design with positive semi definite lyapunov functions abstract a useful stability analysis technique from continuous time nonlinear systems iggidr et al systems control lett 9 1996 95 is extended to the discrete time domain the result is illustrated on a practical example

isolated singularities of solutions to quasi linear elliptic equations with absorption we study the problem of removability of isolated singularities for a general second order quasi linear equation in divergence form diva x u u a0 x u g x u 0 in a punctured domain 0 where is a domain in rn n 3 the model example is the equation pu gu u p 2 u u q 1 0 q p 1 0 p n assuming that the lower order terms satisfy certain non linear kato type conditions we prove that for q n p 1 n p all point singularities of the above equation are removable thus extending the seminal result of brezis and veron

positive radial solutions for quasilinear biharmonic equations abstract we study the quasilinear biharmonic problem u p 2 u f x u g x x b 1 u u 0 x b 1 where b 1 x r n x 1 with n 2 criteria are established for the existence of at least one positive radial solution of the problem explicit lower and upper bounds of the solutions are obtained in our results

on a singular elliptic system with quadratic growth in the gradient abstract in this paper we deal with the following singular elliptic system u u 2 u p p q a x v q u p 2 u f x r n v v 2 v q p q a x u p v q 2 v g x r n where n 3 n 2 4 p q 1 and p q n 2 n 2 we show through the sub and supersolutions method the existence of a nonnegative solution for an approximated system the limit of the approximated solution is a positive solution in the case 0 p q and f g we prove the uniqueness of a solution among others we prove some existence and uniqueness results for some auxiliary problems by using the comparison principle a minimization method and with the help of nehari manifold the proofs rely on the concentration compactness principle

nonexistence of nonnegative solutions of elliptic systems of divergence type abstract in this paper we deal with noncoercive elliptic systems of divergence type that include both the p laplacian and the mean curvature operator and whose right hand sides depend also on a gradient factor we prove that any nonnegative entire weak solution is necessarily constant the main argument of our proofs is based on previous estimates given in filippucci 2009 12 for elliptic inequalities actually the main technique for proving the central estimate has been developed by mitidieri and pohozaev 2001 23 and relies on the method of test functions no use of comparison and maximum principles or assumptions on symmetry or behavior at infinity of the solutions are required

on positive solutions of critical schrodinger operators in two dimensions abstract using a probabilistic approach based on the feynman kac formalism and the spectral radius of the shuttle operator we prove that two dimensional schrodinger operators h v with short range potentials v satisfying v x x 0 x 2 ln x 2 for some 0 are critical if and only if h 0 has a positive bounded distributional solution it is shown that apart from logarithmetic refinements our decay assumptions on v x as x are the best possible this yields a complete solution of a problem posed by simon and extends an earlier result of murata

positive solutions of a class of second order semilinear elliptic equations in the plane on considere l equation elliptique a 2 dimensions u f x u u x r 2 ou x x 1 x 2 et f x u p est une fonction continue non negative sur r 2 r xr 2 on considere l existence de solutions positives definies dans tout r 2

classification of minimal actions of a compact kac algebra with amenable dual we show the uniqueness of minimal actions of a compact kac algebra with amenable dual on the afd factor of type ii1 this particularly implies the uniqueness of minimal actions of a compact group our main tools are a rohlin type theorem the 2 cohomology vanishing theorem and the evans kishimoto type intertwining argument

on positive radial entire solutions of second order quasilinear elliptic systems abstract we study positive radial entire solutions of second order quasilinear elliptic systems of the form p u f x u v q v g x u v x r n sufficient and or necessary conditions of f and g are obtained for to have positive radial entire solutions with prescribed asymptotic behavior at infinity we first study the upper and lower estimates for the nonlinear integral operator which is inverse of one dimensional polar form of the n dimensional m laplace operator

multiplicity of positive radially symmetric solutions for a quasilinear biharmonic equation in the plane abstract this paper is concerned with the multiplicity of positive radially symmetric solutions of the dirichlet boundary value problem for the following two dimensional quasilinear biharmonic equation u p 2 u g x f u x b 1 where b 1 is the unit ball in the plane we apply the fixed point index theory and the upper and lower solutions method to investigate the multiplicity of positive radially symmetric solutions we have found that there exists a threshold such that if then the problem has no positive radially symmetric solution while if 0 then the problem admits at least one positive radially symmetric solution especially there exist at least two positive radially symmetric solutions for 0

stability regions of nonlinear autonomous dynamical systems a topological and dynamical characterization of the stability boundaries for a fairly large class of nonlinear autonomous dynamic systems is presented the stability boundary of a stable equilibrium point is shown to consist of the stable manifolds of all the equilibrium points and or closed orbits on the stability boundary several necessary and sufficient conditions are derived to determine whether a given equilibrium point or closed orbit is on the stability boundary a method for finding the stability region on the basis of these results is proposed the method when feasible will find the exact stability region rather than a subset of it as in the lyapunov theory approach several examples are given to illustrate the theoretical prediction

three solutions for a class of quasilinear elliptic systems involving the p q laplacian abstract the existence of at least three weak solutions is established for a class of quasilinear elliptic systems involving the p q laplacian with dirichlet boundary condition our technical approach is based on the three critical points theorem obtained by b ricceri

unbounded positive entire solutions of semilinear elliptic equations on etudie l existence et le comportement asymptotique des solutions positives non bornees d equations elliptiques semilineaires du type u f x u dans r n on etablit des resultats globaux

on the removability of isolated singular points for degenerating nonlinear elliptic equations abstract the well known results on the removable singularity of elliptic equations are generalized to the class of degenerating nonlinear elliptic equations a sufficient condition for the isolated singular point to be removable has been found in the absence of degeneration this condition coincides with already known results

compact abelian prime actions on von neumann algebras we classify the compact abelian actions on semifinite injective von neumann algebras with factor fixed point algebra the method uses that the nonzero eigenspaces of such an action contain unitaries which give rise to a classifying invariant

criticality theory of half linear equations with the p a laplacian abstract we study positive solutions of half linear second order elliptic equations of the form q a v u div u a p 2 a x u v x u p 2 u 0 in where 1 p is a domain in r n n 2 v l loc a a i j l loc r n 2 is a symmetric and locally uniformly positive definite matrix in and a 2 a x i j 1 n a i j x i j x 1 n r n we extend criticality theory which has been established for linear operators and for half linear operators involving the p laplacian to the operator q a v we prove liouville type theorems and study the behavior of positive solutions of the equation q a v u 0 near an isolated singularity and near infinity in and obtain some perturbations results

remarks on nonexistence of solutions of mixed sublinear and superlinear elliptic equation in rn abstract in this note we consider the quasilinear elliptic equation p u h x u m h x u n in r n n 3 where 0 m p 1 n we establish conditions sufficient to ensure the nonexistence of nonnegative entire solutions which improves a result of a v lair a v lair large solutions of mixed sublinear superlinear elliptic equations j math anal appl 346 2008 99 106

an invariance principle for nonlinear discrete autonomous dynamical systems this note proposes an extension of lasalle s invariance principle for nonlinear discrete autonomous dynamical systems the invariance principle is extended to allow the first difference of the auxiliar scalar function usually a lyapunov function to be positive in some bounded regions moreover a uniform version is proposed to deal with nonlinear discrete dynamical systems that vary with parameters both extensions have the original invariance principle as a particular case as a consequence a larger class of systems can be treated with this new theory the extensions are very useful to obtain attractor estimates as well as their corresponding stability regions the uniform version in particular is useful to obtain estimates that are uniform regarding parameters

existence of entire explosive positive solutions of quasi linear elliptic equations in this paper our main purpose is to establish the existence of entire explosive positive radial solutions for quasi linear elliptic equations the main results of the paper extend previously known results both for the super linear and sub linear cases

radial entire solutions of a class of quasilinear elliptic equations will be examined under suitable conditions on the functions f 8 x r r and g r r where r o 00 r 0 co and 2 is a real parameter the capillarity equation and the equation of prescribed mean curvature are important special cases of 1 1 for which g p 1 p and fir u is suitably specialized 1 2 3 4 10 11 12 131 an entire solution of 1 1 is defined to be a function u e c r satisfying 1 1 at every point xe rn our primary objective is to obtain sufficient conditions on f and g for 1 1 to have positive radial entire solutions of the following three types

stability regions of nonlinear dynamical systems a constructive methodology a constructive methodology for estimating stability regions of general nonlinear dynamical systems is developed the constructive methodology starts with a given lyapunov function either a global or a local lyapunov function and yields a sequence of lyapunov functions which are then used to estimate the stability region the resulting sequence of estimated stability regions is shown to be a strictly monotonic increasing sequence and yet each of them remains inside the entire stability region the significance of this methodology includes its ability to reduce significantly the conservativeness in estimating the stability regions its computational efficiency its adaptability and its sound theoretical basis

some new results on the existence of bounded positive entire solutions for quasilinear elliptic equations in this paper our main purpose is to establish the existence of positive bounded entire solutions of second order quasilinear elliptic equations under new conditions the main results of the present paper are new and extend the previously known results

bounded positive entire solutions of singular quasilinear elliptic equations abstract suppose that 0 is a constant and that f r n r r n r is a continuous function with r 0 this paper investigates n dimensional singular quasilinear elliptic equations of the form u f x u u u 0 x r n n 3 and gives some sufficient conditions for the equations to have i a decaying positive entire solution or ii infinitely many positive entire solutions each of which is bounded with a positive lower bound

nonexistence theorems for singular solutions of quasilinear partial differential equations on etudie les etats fondamentaux singuliers de l equation de poisson semilineaire u f u 0 et de son analogue quasi lineaire div a du du f u 0 ou est le laplacien a n dimensions et a p et f u sont des fonctions donnees

positive solutions of nonlinear p laplacian equations in rn abstract this paper studies the existence of positive solutions of the nonlinear p laplacian equation p u f x u in r n existence of infinitely many positive solutions is established via super sub solution methods decaying conditions on f x u at x are imposed necessity of the conditions and nonexistence of positive solutions are illustrated through decay growth estimates of solutions at infinity special situation and examples are checked in detail

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

hiv dynamics analysis and robust multirate mpc based treatment schedules abstract analysis and control of human immunodeficiency virus hiv infection have attracted the interests of mathematicians and control engineers during the recent years several mathematical models exist and adequately explain the interaction of the hiv infection and the immune system up to the stage of clinical latency as well as viral suppression and immune system recovery after treatment therapy however none of these models can completely exhibit all that is observed clinically and account the full course of infection besides model inaccuracies that hiv models suffer from some disturbances uncertainties from different sources may arise in the modelling in this paper we study the basic properties of a 6 dimensional hiv model that describes the interaction of hiv with two target cells cd4 t cells and macrophages the disturbances are modelled in the hiv model as additive bounded disturbances highly active antiretroviral therapy haart is used the control input is defined to be dependent on the drug dose and drug efficiency we developed treatment schedules for hiv infected patients by using robust multirate model predictive control mpc based method the mpc is constructed on the basis of the approximate discrete time model of the nominal model we established a set of conditions which guarantee that the multirate mpc practically stabilizes the exact discrete time model with disturbances the proposed method is applied to the stabilization of the uninfected steady state of the hiv model the results of simulations show that after initiation of haart with a strong dosage the viral load drops quickly and it can be kept under a suitable level with mild dosage of haart moreover the immune system is recovered with some fluctuations due to the presence of disturbances

the non commutative flow of weights on a von neumann algebra abstract the flow of weights of connes and takesaki is a canonical functor from the category of separable factors to the category of ergodic flows the non commutative flow of weights is another canonical functor from the category of separable factors to the category of covariant systems of semi finite von neumann algebras equipped with trace scaling one parameter automorphism groups with conjugations as morphisms the constructions of these two functors are very similar the flow of weights functor is obtained by looking at all semi finite normal weights on a factor with the murray von neumann equivalence relation the non commutative flow of weights functor is obtained by relating an arbitrary pair of faithful semi finite normal weights by the connes cocycle not only does this construction put a period to the search for a canonical construction of the core of a factor of type iii but it also allows us to put the characteristic square of a factor obtained by katayama sutherland and takesaki in a new perspective the power of this new approach is seen in an ultimate solution to a long standing question of extending the extended modular automorphism of a dominant weight to an arbitrary weight which has been left open ever since the introduction of extended modular automorphisms by connes and takesaki over 20 years ago the construction of the functor ties together the theory of l p spaces of haagerup kosaki hilsum terp and izumi to the structure theory of a factor of type iii in fact the non commutative flow of weights is obtained by the analytic continuation of l p spaces to a pure imaginary value of p

multiplicity theorems for a class of dirichlet quasilinear elliptic systems involving the p1 pn laplacian abstract in this paper we prove the existence of at least three weak solutions for the quasilinear elliptic systems p 1 u 1 f u 1 x u 1 u 2 u n 0 in p 2 u 2 f u 2 x u 1 u 2 u n 0 in p n u n f u n x u 1 u 2 u n 0 in u i 0 for 1 i n on our main tool is a recent three critical points theorem of averna and bonanno d averna g bonanno a three critical points theorem and its applications to the ordinary dirichlet problem topol methods nonlinear anal 22 2003 93 104

bounded positive entire solutions of singular p laplacian equations abstract we investigate singular p laplacian equations of the form div u p 2 u f x u u u 0 x r n where 1 p n 0 p 1 and give a sufficient condition for the equation to have a decaying positive entire solution by means of a supersolution and a subsolution

existence of positive entire solutions for singular and non singular quasi linear elliptic equation we consider the quasilinear elliptic equation div u p 2 u m x u 0 in domain rn where 0 p 1 under several hypotheses on the m x we obtain the existence of positive entire solutions

the structure of the automorphism group of an injective factor and the cocycle conjugacy of discrete abelian group actions we complete classification of discrete abelian or finite group actions on injective type iii1 factors up to cocycle conjugacy we also give a proof for connes characterization of the ker mod and cnt m for an injective factor m of type iii 0 introduction the purpose of this paper is to give a proof of connes announcement on approximately inner automorphisms and centrally trivial automorphisms of an injective 1980 mathematics subject classification 1985 revision 46l40

smooth lyapunov functions and robustness of stability for difference inclusions we demonstrate the equivalence of robust global asymptotic stability gas of the origin and the existence of a smooth lyapunov function for difference inclusions defined by upper semicontinuous set valued maps sufficient conditions for robust gas are given as an application of these results we give conditions for robust gas of difference equations defined by discontinuous right hand sides

existence of entire large positive solutions of semilinear elliptic systems abstract we show that entire large positive radial solutions exist for the semilinear elliptic system u p x v v q x u on r n n 3 for positive and provided that the nonnegative functions p and q are continuous c positive and satisfy the decay conditions 0 tp t dt 0 tq t dt and greater than unity and 0 tp t dt and 0 tq t dt if neither nor is greater than one

brief paper a discrete time observer design for spacecraft attitude determination using an orthogonality preserving algorithm this paper studies a nonlinear discrete time partial state observer design problem for rigid spacecraft systems particularly for spacecraft attitude estimation the construction is inspired by an orthogonality preserving numerical algorithm and it yields a semiglobal practical asymptotic observer we show that the separation principle holds and hence stabilization using output feedback is possible an example is provided to illustrate the usefulness of our results in a sampled data implementation of an attitude control for a rigid spacecraft using output feedback and to illustrate the advantages of the proposed observer design especially compared to a sample and hold version of a similar type observer designed in continuous time

on uniform global asymptotic stability of nonlinear discrete time systems with applications this paper presents new characterizations of uniform global asymptotic stability for nonlinear and time varying discrete time systems under mild assumptions it is shown that weak zero state detectability is equivalent to uniform global asymptotic stability for globally uniformly stable systems by employing the notion of reduced limiting systems another characterization of uniform global asymptotic stability is proposed on the basis of the detectability for the reduced limiting systems associated with the original system as a by product we derive a generalized discrete time version of the well known krasovskii lasalle theorem for general time varying not necessarily periodic systems furthermore we apply the obtained stability results to analyze uniform asymptotic stability of cascaded time varying systems and show that some technical assumptions in recent papers can be relaxed through a practical application it is shown that our results play a similar role to the classic lasalle invariance principle in guaranteeing attractivity noting that reduced limiting systems are used instead of the original system to validate the conceptual characterizations we study the problem of sampled data stabilization for the benchmark example of nonholonomic mobile robots via the exact discrete time model rather than approximate models this case study also reveals that in general sampled data systems may become non periodic even though their original continuous time system is periodic a novel sampled data stabilizer design is proposed using the new stability results and is supported via simulation results

existence of three solutions for a class of dirichlet quasilinear elliptic systems involving the p1 pn laplacian abstract in this paper we prove the existence of at least three weak solutions for the quasilinear elliptic systems p 1 u 1 f u 1 x u 1 u 2 u n 0 in p 2 u 2 f u 2 x u 1 u 2 u n 0 in p n u n f u n x u 1 u 2 u n 0 in u i 0 for 1 i n on our main tool is a recent three critical points theorem of ricceri b ricceri on a three critical points theorem arch math basel 75 2000 220 226

on the existence of radial positive entire solutions for polyharmonic systems abstract two dimensional nonlinear polyharmonic systems of the type n u j p j 2 n u j f j x u 1 u 2 u 1 u 2 x r 2 j 1 2 are considered where p j 1 is a constant and f j 0 0 2 0 2 0 is a continuous function for j 1 2 some sufficient conditions are obtained for the existence of infinitely many radial positive entire solutions of the system with the prescribed asymptotic behavior at infinity

positive entire solutions for singular p laplacian equations on rn with p n 2 abstract suppose that constants p n 2 0 and that f r n r r n r is a continuous function with r 0 this paper discusses the existence of the positive entire solutions of the singular p laplacian equation div u p 2 u f x u u u x r n and gives some sufficient conditions for the equation to have infinitely many positive entire solutions u x satisfying c 1 x u x c 2 x for x 1 where c 1 c 1 0 are constants depending only on u x max 1 log x for p n and x x p n p 1 for p n the super subsolution method is used to prove the existence of such solutions

a new discrete time robust stability condition a new robust stability condition for uncertain discrete time systems with convex polytopic uncertainty is given it enables to check stability using parameter dependent lyapunov functions which are derived from lmi conditions it is shown that this new condition provides better results than the classical quadratic stability besides the use of a parameter dependent lyapunov function this condition exhibits a kind of decoupling between the lyapunov and the system matrices which may be explored for control synthesis purposes a numerical example illustrates the results

existence of positive bounded entire solutions for quasilinear elliptic equations in this paper our main purpose is to establish the existence of positive bounded entire solutions of second order quasilinear elliptic equations the main results of the present paper are new and extend the previously known results

existence and nonexistence of entire positive solutions of semilinear elliptic systems abstract we show that entire positive solutions exist for the semilinear elliptic system u p x v v q x u on r n n 3 for positive and provided that the nonnegative functions p and q are continuous and satisfy appropriate decay conditions at infinity we also show that entire solutions fail to exist if the functions p and q are of slow decay

existence of explosive positive solutions of quasilinear elliptic equations abstract in this paper our main purpose is to consider the quasilinear equation div u p 2 u m x f u on a domain r n n 3 where f is a nonnegative nondecreasing continuous function which vanishes at the origin and m is a nonnegative continuous function with the property that any zero of m is contained in a bounded domain in such that m is positive on its boundary for bounded we show that a nonnegative solution u satisfying u x as x exists for un boundary including r n we show that a similar result holds where u x as x within and u x as x

existence of nontrivial solutions for singular quasilinear elliptic equations on r n in this paper we study the existence of weak solutions for the singular quasilinear elliptic problem 0 1 div x a p u p 2 u v x u p 2u h x u s 2u h x u r 2u x r n u x 0 as x where 1

isolated singularities of positive solutions of p laplacian type equations in r d we study the behavior of positive solutions of p laplacian type elliptic equations of the form q u p u v u p 2u 0in near an isolated singular point where 1 1 we obtain removable singularity theorems for positive solutions near in particular using a new three spheres theorems for certain solutions of the above equation near we prove that if v belongs to a certain kato class near and p d respectively p d then any positive solution u of the equation q u 0 in a punctured neighborhood of 0 respectively is in fact continuous at under further assumptions we find the asymptotic behavior of u near

positive entire solutions to nonlinear biharmonic equations in the plane abstract two dimensional nonlinear biharmonic equations of the form u p 2 u f x u x r2 are considered where p 1 is a constant and f r r r is a continuous function it can be shown that any positive radially symmetric solution is unbounded and grows at least as fast as positive constant multiples of x 2 log x 1 p 1 as x in this paper sharp conditions on f are presented under which has infinitely many positive symmetric entire solutions which are asymptotic to positive constant multiples of x 2 log x 1 p 1 as x

existence and nonexistence results for critical growth polyharmonic elliptic systems abstract in this work we consider semilinear elliptic systems for the polyharmonic operator having a critical growth nonlinearity we establish conditions for existence and nonexistence of nontrivial solutions to these systems

radial entire solutions to even order semilinear elliptic equations in the plane semilinear elliptic equations of the form are considered where m is the m th iterate of the two dimensional laplacian p t is continuous in 0 and f u is continuous and positive either in 0 or in our main objective is to present conditions on p and f which imply the existence of radial entire solutions to that is those functions of class c 2 m 2 which depend only on x and satisfy pointwise in 2 first necessary and sufficient conditions are established for equation with p t 0 in 0 to possess infinitely many positive radial entire solutions which are asymptotic to positive constant multiples of x 2 m 2 log x as x secondly it is shown that in the case p t f u 0 is nondecreasing in equation always has eventually negative radial entire solutions all of which decrease at least as fast as negative constant multiples of x 2 m 2 log x as x our results seem to be new even when specialised to the prototypes where is a constant

existence and nonexistence of solutions for quasilinear elliptic equations abstract in this paper we study the problem p u f x u u in u 0 on where r n is a smooth bounded domain n 2 and p u div u p 2 u defines the p laplacian we provide some necessary and sufficient conditions on f under which the problem admits a weak solution for the case p 2 we obtain more general conditions on f the main ingredients are degree theory and the super subsolution method

a remark on the existence of entire solutions of a singular semilinear elliptic problem it is proved that the singular semilinear elliptic equation u p x g u 0 p x x rn 2 n lims 0 g s andg c1 0 0 which is strictly decreasing in 0 has a unique positivec2 loc rn solution that decays to zero near provided 0 t t dt where t imax i x tg x

survey paper set invariance in control the properties of positively invariant sets are involved in many different problems in control theory such as constrained control robustness analysis synthesis and optimization in this paper we provide an overview of the literature concerning positively invariant sets and their application to the analysis and synthesis of control systems

on the behaviour of harmonic functions near isolated points abstract in this work we study the behaviour of harmonic functions near isolated exceptional points we discuss two situations i when points are in the interior and ii when points are on a flat portion of the boundary on which a solution vanishes we consider both bounded and unbounded solutions under the assumptions of boundedness solutions are well behaved in either case we show that unbounded solutions in i change sign and provide some lower bounds for growth rates valid for both i and ii in ii if the solution has one sign then we can also find an upper bound we also look at the case of the ball

existence of three solutions for a class of quasilinear elliptic systems involving the p x q x laplacian abstract we study the solutions of the p x q x laplacian equations with dirichlet boundary condition on a bounded domain and obtain three solutions under appropriate hypotheses our technical approach is based on the general three critical points theorem obtained by b ricceri