on lp x norms the relation between a banach function space x and its subspaces formed by i those functions with absolutely continuous norm ii those with continuous norm and iii the closure of the set of

positive forms and stability of linear time delay systems we consider the problem of constructing lyapunov functions for linear differential equations with delays for such systems it is known that exponential stability implies the existence of a positive lyapunov function which is quadratic on the space of continuous functions we give an explicit parameterization of a sequence of finite dimensional subsets of the cone of positive lyapunov functions using positive semidefinite matrices this allows stability analysis of linear time delay systems to be formulated as a semidefinite program

delay induced instabilities in gyroscopic systems it is shown that stable linear gyroscopic systems of the form mx tx kx 0 m 0 always become unstable when an arbitrarily small delay is introduced in the gyroscopic term in the case where k is negative definite then the system will be unstable for all positive delays on the other hand examples are given showing that some of these systems may actually become asymptotically stable for larger values of the delay parameter

multiple solutions for a class of p x laplacian equations in involving the critical exponent in this paper we first establish a principle of concentration compactness in then based on this concentration compactness principle we study the existence of solutions for a class of p x lapl

multiple positive solutions of superlinear elliptic problems with sign changing weight we study the existence of multiple positive solutions for a superlinear elliptic pde with a sign changing weight our approach is variational and relies on classical critical point theory on smooth manifolds a special care is paid to the localization of minimax critical points

embedded area constrained willmore tori of small area in riemannian three manifolds ii morse theory the synthesis of cationic rhodium and iridium complexes of a bis imidazol 2 thione functionalised calix 4 arene ligand and their surprising capacity for potassium binding is described in both cases uptake of the alkali metal into the calix 4 arene cavity occurs despite adverse electrostatic interactions associated with close proximity to the transition metal fragment rh k 3 715 1 a ir k 3 690 1 a the formation and constituent bonding of these unusual heterobimetallic adducts has been interrogated through extensive solution and solid state characterisation examination of the host guest chemistry of the ligand and its upper rim unfunctionalised calix 4 arene analogue and computationally using dft based energy decomposition analysis eda this is the second part of a series of two papers where we construct embedded willmore tori with small area constraint in riemannian three manifolds in both papers the construction relies on a lyapunov schmidt reduction the difficulty being the moebius degeneration of the tori in the first paper the construction was performed via minimization here by morse theory to this aim we establish new geometric expansions of the derivative of the willmore functional on small clifford tori in geodesic normal coordinates which degenerate to small geodesic spheres with a small handle under the action of the moebius group by using these sharp asymptotics we give sufficient conditions in terms of the ambient curvature tensors and morse inequalities for having existence multiplicity of embedded tori which are stationary for the willmore functional under the constraint of prescribed sufficiently small area

a multiplicity result for singular nls equations with magnetic potentials abstract in this paper we obtain multiple solutions u r n c of the singular nonlinear schrodinger equation with an external electromagnetic field 1 i a x 2 u v x e w x u u p 2 u x r n where n 2 2 p 2 a is a c 1 real valued vector magnetic potential v x is a real continuous lower bounded electric potential and e w x is a perturbing potential depending on which may be singular our result covers the meaningful physical cases w x 1 x w x 1 x 2 we prove that the number of solutions to the eq 1 is bounded below by some number which depends on the category of the set of global minima of v for sufficiently small

numerical optimization for feedback stabilization of linear systems with distributed delays abstract this paper investigates the feedback stabilization of linear systems with distributed delays by means of the state transition matrix fundamental matrix of the delay systems an optimization problem is formulated for solving the feedback stabilization problem we prove a necessary and sufficient condition for feedback gain matrices to stabilize the delay systems then a numerical method based on the condition is presented to determine the feedback gain matrices furthermore numerical examples show that the presented algorithm is efficient for stabilization of the delay systems

multiplicity results for some nonlinear schrodinger equations with potentials multiple semiclassical standing waves for a class of nonlinear schrodinger equations with potentials are found by means of a perturbative variational method

complete quadratic lyapunov functionals for distributed delay systems this paper is concerned with the stability analysis of distributed delay systems using complete lyapunov functionals numerous articles aim at approximating their parameters thanks to a discretization method or polynomial modeling the interest of such approximations is the design of tractable sufficient stability conditions in the present article we provide an alternative method based on polynomial approximation which takes advantages of the legendre polynomials and their properties the resulting stability conditions are scalable with respect to the maximum degree of the polynomials and are expressed in terms of tractable linear matrix inequalities several examples of delayed systems are tested to show the effectiveness of the method

homoclinic solutions of an infinite dimensional hamiltonian system we consider the system begin equation left begin array l partial t u delta x u v x u h v t x u v 3mm partial t v delta x v v x v h u t x u v end array qquad text for t x in mathbb r times mathbb r n right end equation which is an unbounded hamiltonian system in l 2 mathbb r n mathbb r 2m we assume that the constant function u 0 v 0 equiv 0 0 in mathbb r 2m is a stationary solution and that h and v are periodic in the t and x variables we present a variational formulation in order to obtain homoclinic solutions z u v satisfying z t x to 0 as t x to infty it is allowed that v changes sign and that delta v has essential spectrum below and above 0 we also treat the case of a bounded domain omega instead of mathbb r n with dirichlet boundary conditions

generation of tetrahedral mesh of variable element size by sphere packing over an unbounded 3d domain this paper describes an algorithm for the generation of tetrahedral mesh of specified element size over an unbounded three dimensional domain starting from an arbitrary point in space defined as the origin and guided by the concept of advancing front spheres of size compatible with the specified element size are packed tightly together one by one to form a cluster of spheres of different sizes the compactness of the cluster of spheres is achieved by packing spheres at a site closest to the origin in a densest manner with tangent but no overlapping with as many other spheres as possible in view of these criteria a rotational mechanism between spheres is innovated which allows the newly inserted sphere to follow the path by rotation between existing spheres until the lowest point is reached the centres of the packing spheres provide ideal locations for delaunay point insertion to form a triangulation of tetrahedral elements of size compatible with the specified value spheres of random size distribution or of size specified by a node spacing function are packed by the proposed algorithm in all test examples high quality tetrahedral elements of size consistent with the specified node spacing are generated the process is fast and robust and the time complexity for mesh generation is expected to be almost linear however in the present implementation a quasi linear time relationship is observed as a search for the nearest node on the front is proposed in the sphere packing process the memory requirement has been kept to a minimum as no additional data structure other than the adjacency relationship of the tetrahedral elements as required by the delaunay triangulation is stored the structure of the advancing frontal surface needs not be explicitly constructed nor updated as it is simply not required in the sphere packing process

discretized lyapunov functional for systems with distributed delay and piecewise constant coefficients the stability problem for systems with distributed delay is considered using discretized lyapunov functional the coefficients associated with the distributed delay are assumed to be piecewise constant and the discretization mesh may be non uniform the resulting stability criteria are written in the form of linear matrix inequality numerical examples are also provided to illustrate the effectiveness of the method the basic idea can be extended to a more general setting with more involved formulation

embedded area constrained willmore tori of small area in riemannian three manifolds i minimization we construct embedded willmore tori with small area constraint in riemannian three manifolds under some curvature condition used to prevent mobius degeneration the construction relies on a lyapunov schmidt reduction to this aim we establish new geometric expansions of exponentiated small symmetric clifford tori and analyze the sharp asymptotic behaviour of degenerating tori under the action of the mobius group in this first work we prove two existence results by minimizing or maximizing a suitable reduced functional in particular we obtain embedded area constrained willmore tori or equivalently toroidal critical points of the hawking mass under area constraint in compact 3 manifolds with constant scalar curvature and in the double schwarzschild space in a forthcoming paper new existence theorems will be achieved via morse theory

stability of linear systems with time varying delays using bessel legendre inequalities this paper addresses the stability problem of linear systems with a time varying delay hierarchical stability conditions based on linear matrix inequalities are obtained from an extensive use of the bessel inequality applied to legendre polynomials of arbitrary orders while this inequality has been only used for constant discrete and distributed delays this paper generalizes the same methodology to time varying delays we take advantages of the dependence of the stability criteria on both the delay and its derivative to propose a new definition of allowable delay sets a light and smart modification in their definition leads to relevant conclusions on the numerical results

a multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids the goal of this correction is to correct a mistake that appears in the proof of both lemma 3 9 and 3 10 in the article mih ilescu r dulescu 2006 1 the mistake in the proof of lemma 3 9 and 3 10 is mainly owing to the fact that we wrongly computed the expression of function g x t

existence and multiplicity of solutions for p x laplacian equations in rn abstract this paper investigates the existence and multiplicity of solutions for p x laplacian equations in r n

foliations of small tubes in riemannian manifolds by capillary minimal discs abstract letting be an embedded curve in a riemannian manifold m we prove the existence of minimal disc type surfaces centered at inside the surface of revolution of m around having small radius and intersecting it with constant angles in particular we obtain that small tubular neighborhoods can be foliated by minimal discs

stabilization of uncertain linear distributed delay systems with dissipativity constraints abstract this paper examines the problem of stabilizing linear distributed delay systems with nonlinear distributed delay kernels and dissipativity constraints specifically the nonlinear distributed kernel includes functions such as polynomials trigonometric and exponential functions by constructing a liapunov krasovskii functional related to the distributed kernels sufficient conditions for the existence of a state feedback controller which stabilizes the uncertain distributed delay systems with dissipativity constraints are given in terms of linear matrix inequalities lmis in contrast to existing methods the proposed scenario is less conservative or requiring less number of decision variables based on the application of a new derived integral inequality finally numerical examples are presented to demonstrate the validity and effectiveness of the proposed methodology

an overlapping detection algorithm for random sequential packing of elliptical particles abstract random sequential packing of particles is a subject of intense research in many branches of physics and engineering the preponderance of preview work has focused on spherical particles and little is known about non overlapping elliptical particles in the present work a new numerical algorithm is developed to detect overlapping of ellipses by a series of sequential coordinate transformations and a novel golden section search algorithm the accuracy and efficiency of the numerical algorithm are tested and compared with algorithms developed in previous literature its stability is verified by visualizations of random sequential packing of monodispersed and polydispersed ellipses with different boundary conditions then the algorithm is applied to investigate the influence of the shape of ellipses on the random packing fraction as well as to study the influence of the shape and size of ellipses on the wall effect in the particle packing structure and some numerical results are demonstrated finally the reliability of the numerical algorithm extended to the three dimensional space is also evaluated by checking overlapping of ellipsoidal particles

existence of positive solutions of some nonlinear elliptic problems in unbounded domains in this paper we study the existence of positive solutions of some nonlinear elliptic problems in unbounded domains the existence is affected by the properties of the geometry and the topology of the domain

a palais smale approach to lane emden equations we consider the unbounded domain problems u u u p 2 u in u 0i n a ndu 0o n w here is an unbounded domain in r n 2 2 and 2 for n 2 the existence of a ground state solution to the problems is greatly affected by the shape of the domain to determine the existence of the solutions in a general domain remains a challenge task for the flat interior flask domain that consists a strip and a ball attached to the bottom of the strip previous results have asserted the existence of a ground state solution when the diameter of the ball is greater than a positive constant however the existence of the solutions when the diameter of the ball equals to the width of the strip is still an important open question this article resolves the open question partially by considering a variation of the flat interior flask domain which is formed by attaching a stretched ball to the bottom of the strip

the principle of concentration compactness in lp x spaces and its application abstract in this paper we establish a principle of concentration compactness in l p x spaces and apply it to obtain the existence of solutions for p x laplacian equations with critical growth

multibump nodal solutions for an indefinite superlinear elliptic problem abstract we define some nehari type constraints using an orthogonal decomposition of the sobolev space h 0 1 and prove the existence of multibump nodal solutions for an indefinite superlinear elliptic problem

monte carlo study of the sphere packing problem we employ the monte carlo method to study a constrained optimization problem that is packing spheres with unequal radii into a 3 d bounded region selection of the best fit solution is based on using the boltzmann factor e e t to determine the transition probability which allows us to search for the global optimal solution we determined the least numbers of packed spheres that will occupy the largest volume the optimal occupied volume found is around 44 of a bounded region volume which is obtained within a relative short computing time this suggests that our result could be able to give a good starting point for the radiosurgery treatment plan

on a class of singularly perturbed elliptic equations in divergence form existence and multiplicity results the main purpose of this paper is to study the existence of single peaked positive solutions of the singularly perturbed elliptic equation e2div j x u v x u f u inrn where j is a symmetric uniformly elliptic matrix and v is a positive potential possibly unbounded from above if f u up then solutions concentrate at non degenerate critical points of x v x p 1p 1 n2 detj x 1 2

positive solutions for a weakly coupled nonlinear schrodinger system abstract existence of a nontrivial solution is established via variational methods for a system of weakly coupled nonlinear schrodinger equations the main goal is to obtain a positive solution of minimal action if possible with all vector components not identically zero generalizations for nonautonomous systems are considered

perturbation results of critical elliptic equations of caffarelli kohn nirenberg type abstract we find for small e positive solutions to the equation div x 2a u x 2 1 a u 1 ek x u p 1 x bp in r n which branch off from the manifold of minimizers in the class of radial functions of the corresponding caffarelli kohn nirenberg type inequality moreover our analysis highlights the symmetry breaking phenomenon in these inequalities namely the existence of non radial minimizers

wirtinger based integral inequality application to time delay systems in the last decade the jensen inequality has been intensively used in the context of time delay or sampled data systems since it is an appropriate tool to derive tractable stability conditions expressed in terms of linear matrix inequalities lmis however it is also well known that this inequality introduces an undesirable conservatism in the stability conditions and looking at the literature reducing this gap is a relevant issue and always an open problem in this paper we propose an alternative inequality based on the fourier theory more precisely on the wirtinger inequalities it is shown that this resulting inequality encompasses the jensen one and also leads to tractable lmi conditions in order to illustrate the potential gain of employing this new inequality with respect to the jensen one two applications on time delay and sampled data stability analysis are provided

delay range stability of a class of distributed time delay systems this paper is dedicated to the stability analysis of a class of uncertain distributed delay systems the kernel of which can be modeled as a polynomial function of the delay the results are constructed by rewriting the system as an uncertain interconnected model appropriate robust control tools i e quadratic separation are then used to address the stability issue to this end some relations that highlight relevant characteristics of the delayed term are added to the interconnected model leading then to the conservatism reduction finally numerical examples show the effectiveness of the proposed method

sobolev embedding theorems for spaces wk p x ω this paper gives a sobolev type embedding theorem for the generalized lebesgue sobolev space wk p x where is an open domain in rn n 2 with cone property and p x is a lipschitz continuous function defined on satisfying 1 p p nk the main result can be stated as follows for any measurable function q x x formula with p x q x p x np x n kp x there exists a continuous embedding from wk p x to lq x

h bubbles in a perturbative setting the finite dimensional reduction s method given a regular function h colon mathbb r 3 to mathbb r we look for h bubbles that is regular surfaces in mathbb r 3 parametrized on the sphere mathbb s 2 with mean curvature h at every point here we study the case of h u h 0 varepsilon h 1 u h varepsilon u where h 0 is a nonzero constant varepsilon is the smallness parameter and h 1 is any c 2 function we prove that if bar p in mathbb r 3 is a good stationary point for the melnikov type function gamma p int q p h 0 1 h 1 q dq then for varepsilon small there exists an h varepsilon bubble omega varepsilon that converges to a sphere of radius h 0 1 centered at bar p as varepsilon to 0

multiple positive solutions of semilinear elliptic boundary value problems in the half space abstract in this paper we prove that if q is a positive continuous function in r n and satisfies the suitable conditions then for sufficiently small 0 u u q z u p 1 with the boundary condition u x z n g x admits at least three positive solutions in r n

stability analysis of linear coupled differential difference systems with general distributed delays we present a new approach for the stability analysis of linear coupled differential difference systems with a general distributed delay the distributed delay term in this note can contain any mathbb l 2 function which is approximated via a class of elementary functions including polynomial trigonometric exponential functions etc through the application of a new proposed integral inequality a sufficient condition for the stability of the system is derived in terms of linear matrix inequalities based on the construction of a lyapunov krasovskii functional the methods proposed in this note can handle problems that cannot be dealt using existing approaches two numerical examples are presented to show the effectiveness of our proposed stability condition

three positive solutions of semilinear elliptic equations in the half space with a hole abstract in this paper assume that h is nonnegative and h l 2 0 we prove that if h l 2 is sufficiently small then there are at least three positive solutions of eq 1 in r n d where d is a c 1 1 bounded domain in r n

pseudospectral differencing methods for characteristic roots of delay differential equations in d breda s maset and r vermiglio ima j numer anal 24 2004 pp 1 19 and d breda the infinitesimal generator approach for the computation of characteristic roots for delay differential equations using bdf methods research report udmi rr17 2002 dipartimento di matematica e informatica universita degli studi di udine udine italy 2002 the authors proposed to compute the characteristic roots of delay differential equations ddes with multiple discrete and distributed delays by approximating the derivative in the infinitesimal generator of the solution operator semigroup by runge kutta rk and linear multistep lms methods respectively in this work the same approach is proposed in a new version based on pseudospectral differencing techniques we prove the spectral accuracy convergence behavior typical of pseudospectral schemes as also illustrated by some numerical experiments

bifurcation results for a class of p x laplacian equations abstract in this paper we study the following p x laplacian equation div u p x 2 u u p x 2 u f x u x u x 0 x where r n is bounded 0 under suitable assumptions we obtain the existence of global branches of solutions for the above problem via the subsolution supersolution method

semiclassical limit for nonlinear schrodinger equations with electromagnetic fields abstract we study the existence of standing waves for a class of nonlinear schrodinger equations in r n with both an electric and a magnetic field under suitable non degeneracy assumptions on the critical points of an auxiliary function related to the electric field we prove the existence and the multiplicity of complex valued solutions in the semiclassical limit we show that in the semiclassical limit the presence of a magnetic field produces a phase in the complex wave but it does not influence the location of peaks of the modulus of these waves

hierarchical stability conditions for a class of generalized neural networks with multiple discrete and distributed delays this brief investigates the analysis issue for global asymptotic stability of a class of generalized neural networks with multiple discrete and distributed delays to tackle delays arising in different neuron activation functions we employ a generalized model with multiple discrete and distributed delays which covers various existing neural networks we then generalize the bessel legendre inequalities to deal with integral terms with any linearly independent functions and nonlinear function of states based on these inequalities we design the lyapunov krasovskii functional and derive hierarchical linear matrix inequality stability conditions finally three numerical examples are provided to demonstrate that the proposed method is less conservative with a reasonable numerical burden than the existing results

multiple integral inequalities and stability analysis of time delay systems abstract this paper is devoted to stability analysis of continuous time delay systems based on a set of lyapunov krasovskii functionals new multiple integral inequalities are derived that involve the famous jensen s and wirtinger s inequalities as well as the recently presented bessel legendre inequalities of seuret and gouaisbaut 2015 and the wirtinger based multiple integral inequalities of park et al 2015 and lee et al 2015 the present paper aims at showing that the proposed set of sufficient stability conditions can be arranged into a bidirectional hierarchy of lmis establishing a rigorous theoretical basis for comparison of conservatism of the investigated methods numerical examples illustrate the efficiency of the method

a concentration phenomenon for semilinear elliptic equations for a domain omega subset mathbb r n we consider the equation delta u v x u q n x u p 2 u with zero dirichlet boundary conditions and p in 2 2 here v geqq 0 and qn are bounded functions that are positive in a region contained in omega and negative outside and such that the sets qn 0 shrink to a point x 0 in omega as n to infty we show that if un is a nontrivial solution corresponding to qn then the sequence un concentrates at x0 with respect to the h1 and certain lq norms we also show that if the sets qn 0 shrink to two points and un are ground state solutions then they concentrate at one of these points

a multiplicity result for the yamabe problem on sn abstract we prove a multiplicity result for the yamabe problem on the manifold s g where g is a perturbation of the standard metric g 0 of s n solutions are found by variational methods via an abstract perturbation result

hierarchy of stability criterion for time delay systems based on multiple integral approach abstract taking a class of time delay systems as research object this brief aims at developing a theoretical support on the hierarchy of stability criterion which is derived by the multiple integral approach and free weighting matrix technique the hierarchy implies that the conservatism of stability criterion can be reduced by increasing the ply of integral terms in lyapunov krasovskii functional lkf together with three numerical experiments the hierarchy of stability criterion is further shown

variational perturbative methods and bifurcation of bound states from the essential spectrum this paper consists of two main parts the first deals with a perturbative method in critical point theory and can be seen as the generalisation and completion of some earlier results the second part is concerned with applications of the abstract setup to the existence of bound states of a class of elliptic differential equations that branch off from the infimum of the essential spectrum

symmetry of positive solutions of semilinear elliptic equations in infinite strip domains abstract in this article we apply the improved moving plane method to prove the symmetry of the solutions of the dirichlet problem u u g u in infinite strip domains with zero boundary condition

stability analysis of linear systems with interval time varying delays utilizing multiple integral inequalities this paper is devoted to stability analysis of continuous time delay systems with interval time varying delays having known bounds on the delay derivatives a parameterized family of lyapunov krasovskii functionals involving multiple integral terms is introduced and novel multiple integral inequalities are utilized to derive sufficient stability condition for systems with time varying delays the efficiency of the proposed method is illustrated by numerical examples

a perturbation method for spinorial yamabe type equations on s m and its application for m ge 2 we prove the existence of non trivial solutions for a certain kind of nonlinear dirac equations with critical sobolev nonlinearities on s m via a perturbative variational method for the special case m 2 this establishes the existence of a conformal immersion s 2 rightarrow mathbb r 3 with prescribed mean curvature h which is close to a positive constant under an index counting condition on h

compact imbedding theorems with symmetry of strauss lions type for the space w1 p x ω abstract in this paper we extend some compact imbedding theorems of strauss lions type to the space w 1 p x when the domain has some symmetric properties and p x satisfies some conditions

bifurcation in lp rn for a semilinear elliptic equation keywords lsp p bifurcation constrained minimization sobolev spaces nonlinear eigenvalue problem semilinear spectrum linearisation variational structure reference ana article 1988 001doi 10 1112 plms s3 57 3 511 record created on 2008 12 10 modified on 2016 08 08

varying bifurcation diagrams of positive solutions for a class of indefinite superlinear boundary value problems in this work we analyze the existence multiplicity and stability of positive solutions for a class of indefinite superlinear elliptic boundary value problems the main contribution of this paper consists in the change of mind inherent to the fact of adding the superlinear amplitude e as an unfolding parameter this change of mind allows us to unify many previous results obtained separately in the literature it helps us to realize the global structure of the set of positive steady states and provides us with a great variety of new general results our techniques can be applied to much more general equations and systems

some results about the existence of critical points for the willmore functional using a perturbative approach it is shown existence and multiplicity of critical points for the willmore functional in ambient manifold mathbb r 3 g epsilon where g epsilon is a metric close and asymptotic to the euclidean one with the same technique it is proved a non existence result in general riemannian manifold m g of dimension three

on perturbations of a translationally invariant differential equation on etudie certaines perturbations de l equation differentielle u u up 0 sur tout l espace euclidien a n dimensions on obtient des conditions qui assurent l existence d une solution a l equation perturbee pres d une solution donnee de l equation non perturbee

existence of positive solutions for p x laplacian equations in unbounded domains abstract in this paper we study the existence of positive solutions for p x laplacian equations in unbounded domains the existence is affected by the properties of the geometry and the topology of the domain

bessel laguerre inequality and its application to systems with infinite distributed delays by taking advantage of properties of the laguerre polynomials we propose a new inequality called bessel laguerre integral inequality which can be applied to stability analysis of linear systems with infinite distributed delays and with general kernels the matrix corresponding to the system without the delayed term or the matrix corresponding to the system with the zero delay is not necessarily assumed to be non hurwitz through a laguerre polynomials approximation of kernels the advantage of the method is that the original system is not needed to be transformed into an augmented one instead it is represented as a system with additional signals that are captured by the bessel laguerre integral inequality then we derive a set of sufficient stability conditions that is parameterized by the degree of the polynomials the particular case of gamma kernel functions can be easily considered in this analysis numerical examples illustrate the potential improvements achieved by the presented conditions with increasing the degree of the polynomial but at the price of numerical complexity

exact multiplicity results and bifurcation for nonhomogeneous elliptic problems abstract in this paper we consider the following problem u x u x f x u h x in r n u h 1 r n u 0 in r n where 0 is a parameter we assume lim x f x u f u uniformly on any compact subset of 0 but we do not require f x u f u for all x r n we prove that there exists 0 such that has exactly two positive solutions for 0 no solution for a unique positive solution u for and u is a bifurcation point in c 2 r n w 2 2 r n

existence of multiple positive solutions of semilinear elliptic boundary value problems in the half space abstract in this paper we prove that if q and f satisfy some suitable conditions then u u q x f u in r n with the boundary condition u y 0 g y has at least two positive solutions if 0 a unique positive solution if and no positive solution if

existence of solutions for p x laplacian problems on a bounded domain in this paper we study the following p x laplacian problem div a x vd uv p x 2 d u b x v uv p x 2 u f x u x e u u 0 on p u where 1 p 1 l p x l p 2 n u s r n is a bounded domain and applying the mountain pass theorem we obtain the existence of solutions in w 0 1 p x for the p x laplacian problems in the superlinear and sublinear cases 2004 elsevier inc all rights reserved