on multiple solutions of a semilinear schrodinger equation with periodic potential abstract this paper is concerned with the semilinear schrodinger equation s u v x u f x u u h 1 r n where v and f are periodic in the x variables f is a superlinear and subcritical nonlinearity and 0 lies in a spectral gap of u v it is shown that if f is odd in u then s has infinitely many large energy solutions the proof relies on a generalized variant fountain theorem for strongly indefinite functionals established in this paper

nonlinear schrodinger equations without compatibility conditions on the potentials abstract we study the existence of nonnegative solutions and ground states to nonlinear schrodinger equations in r n with radial potentials and super linear or sub linear nonlinearities the potentials satisfy power type estimates at the origin and at infinity but no compatibility condition is required on their growth or decay rates at zero and infinity in this respect our results extend some well known results in the literature and we also believe that they can highlight the role of the sum of lebesgue spaces in studying nonlinear equations with weights

multiplicity of weak solutions for a class of nonuniformly elliptic equations of p laplacian type abstract this paper deals with the multiplicity of weak solutions in w 0 1 to a class of nonuniformly elliptic equations of the form div a x u h x u r 1 u g x u s 1 u in a bounded domain of r n here a satisfies a x c 0 h 0 x h 1 x p 1 for all r n a e x h 0 l p p 1 h 1 l loc 1 h 1 x 1 for a e x in 1 r p 1 s n p n p n p

emden fowler equations involving critical exponents on considere le probleme y t k y 2 y 2k 3 0 t 0 pour t 2 0 0 soit t inf t 0 y 0 sur t alors t 0 0 on etudie comment la position t du premier zero de y t varie avec

revisit to sign changing solutions for the nonlinear schrodinger poisson system in r3 abstract in this paper we investigate the existence of solutions for the nonlinear schrodinger poisson system with zero mass by introducing some new ideas we prove via the constraint variational method and the quantitative deformation lemma that the system has a sign changing solution

multiple solutions to a linearly coupled elliptic system with critical exponents abstract we are concerned with the linearly coupled elliptic system with critical exponents u a x u u 2 2 u u p 2 u v x r n v b x v v 2 2 v v q 2 v u x r n u v h 1 r n where n 3 2 p q 2 2 n n 2 and are nonnegative constants and a x b x are positive periodic functions the existence of a positive ground state and a higher energy solution of this system is proved using variational methods additionally the asymptotic behavior of these solutions as 0 is obtained

existence of solution for a class of quasilinear schrodinger equation in rn with zero mass abstract in this paper we use variational methods to establish a berestycki lions type result for two classes of quasilinear schrodinger equations in r n in the zero mass situation

generalized fountain theorem and applications to strongly indefinite semilinear problems abstract by using the degree theory and the topology of kryszewski and szulkin we establish a version of the fountain theorem for strongly indefinite functionals the abstract result will be applied for studying the existence of infinitely many solutions of two strongly indefinite semilinear problems including the semilinear schrodinger equation

nonuniformly elliptic equations of p laplacian type abstract this paper deals with the existence of a generalized solution in w 0 1 p to a nonuniformly nonlinear elliptic equation of the form div a x u f x u in a bounded domain of r n here a satisfies a x c 0 h 0 x h 1 x p 1 for any in r n a e x h 0 l p p 1 and h 1 l loc 1

darboux integrability and the inverse integrating factor abstract we mainly study polynomial differential systems of the form dx dt p x y dy dt q x y where p and q are complex polynomials in the dependent complex variables x and y and the independent variable t is either real or complex we assume that the polynomials p and q are relatively prime and that the differential system has a darboux first integral of the form h f 1 1 f p p exp h 1 g 1 n 1 1 exp h q g q n q q where the polynomials f i and g j are irreducible the polynomials g j and h j are coprime and the i and j are complex numbers when i 1 p and j 1 q prelle and singer proved that these systems have a rational integrating factor we improve this result as follows assume that h is a rational function which is not polynomial following to poincare we define the critical remarkable values of h then we prove that the system has a polynomial inverse integrating factor if and only if h has at most two critical remarkable values under some assumptions over the darboux first integral h we show first that the system has a polynomial inverse integrating factor and secondly that if the degree of the system is m the homogeneous part of highest degree of h is a multi valued function and the functions exp h j g j are exponential factors for j 1 q then the system has a polynomial inverse integrating factor of degree m 1 we also present versions of these results for real polynomial differential systems finally we apply these results to real polynomial differential systems having a darboux first integral and limit cycles or foci

positive ground state solutions for the critical klein gordon maxwell system with potentials abstract this paper deals with the klein gordon maxwell system when the nonlinearity exhibits critical growth we prove the existence of positive ground state solutions for this system when a periodic potential v is introduced the method combines the minimization of the corresponding euler lagrange functional on the nehari manifold with the brezis and nirenberg technique

a positive ground state solution for a class of asymptotically periodic schrodinger equations with critical exponent abstract in the paper by using a new condition a class of asymptotically periodic schrodinger equations with critical exponent are studied via the variational method a positive ground state solution is obtained

some new results on darboux integrable differential systems abstract we deal with complex planar differential systems having a darboux first integral h we present a definition of remarkable values and remarkable curves associated to h and characterize the existence of a polynomial inverse integrating factor for these systems furthermore we study the relation between the characteristic polynomial f and the inverse integrating factors of the system and show the importance of the numerator of the exponential factor of h in the construction of f

on the existence of positive solutions of a perturbed hamiltonian system in rn using the legendre fenchel transformation and the mountain pass theorem due to ambrosetti and rabinowitz we establish an existence result for perturbations of periodic and asymptotically periodic semilinear hamiltonian systems of the type pw u u w2 x v p 1vinrn v v w1 x u q 1uinrn u x v x 0as x u 0 v 0inrn n 2 here the numbers p q 1 are below the critical hyperbola if n 3 that is they satisfy 1 p 1 1 q 1 n 2 n while no additional restrictions on p and q are required if n 2 the functions wi i 1 2 are bounded positive continuous functions

multiplicity of solutions for homogeneous elliptic systems with critical growth abstract in this paper we are concerned with the number of nonnegative solutions of the elliptic system u q u u v 1 2 h u u v in v q v u v 1 2 h v u v in u v 0 on where r n is a bounded smooth domain n 4 2 2 n n 2 and q u h u and q v h v are the partial derivatives of the homogeneous functions q h c 1 r 2 r where r 2 0 0 in the proofs we apply variational methods and ljusternik schnirelmann theory

darboux theory of integrability in cn taking into account the multiplicity abstract darboux in 1878 provided a theory on the existence of first integrals of polynomial systems based on the existence of sufficient invariant algebraic hypersurfaces called now the darboux theory of integrability in 1979 jouanolou successfully improved the darboux theory of integrability characterizing the existence of rational first integrals for this he used sophisticated tools of algebraic geometry the aim of this paper is to improve the classical result of darboux and the new one of jouanolou taking into account the multiplicity of the invariant algebraic hypersurfaces additionally our proof of the improved result of jouanolou is much simpler and elementary than the original one some examples show that the improved darboux theory of integrability with multiplicity is much useful than the classical one

on perturbations of a class of a periodic m laplacian equation with critical growth 1 after the well known results of brezis nirenberg 10 problems involving ellipticequationswithcriticalgrowthinboundeddomainshavebeenstudiedbyseveralauthors see for example 7 15 11 13 17 and the references therein the special features of this class of problems considered in this paper is that it isdened in r

existence and concentration of ground state solutions for critical schrodinger equation with steep potential well abstract in this paper we study the following schrodinger equation u 1 a x u f u u 2 2 u in r n where n 3 and 0 we obtain a ground state solution for the above equation moreover the concentration behavior of the ground state solution is also described as

multi bump solutions for a strongly indefinite semilinear schrodinger equation without symmetry or convexity assumptions abstract in this paper we study the following semilinear schrodinger equation with periodic coefficient u v x u f x u u h 1 r n the functional corresponding to this equation possesses strongly indefinite structure the nonlinear term f x t satisfies some superlinear growth conditions and need not be odd or increasing in t using a new variational reduction method and a generalized morse theory we proved that this equation has infinitely many geometrically different solutions furthermore if the solutions of this equation under some energy level are isolated then we can show that this equation has infinitely many m bump solutions for any positive integer m 2

existence of ground state sign changing solutions for a class of generalized quasilinear schrodinger maxwell system in r3 abstract in this paper we study the existence of ground state sign changing solutions for the following generalized quasilinear schrodinger maxwell system d i v g 2 u u g u g u u 2 v x u g u g u k x f u x r 3 g 2 u x r 3 where g c 1 r r v x and k x are positive continuous functions and is a positive parameter by making a change of variable as u g 1 v and g u 0 u g t d t we obtain one ground state sign changing solution v g 1 u by using some new analytical skills and non nehari manifold method furthermore the energy of v g 1 u is strictly larger than twice that of the ground state solutions of nehari type we also establish the convergence property of v g 1 u as the parameter 0 our results improve and generalize some results obtained by chen and tang 2016 zhu et al 2016

on a class of semilinear schrodinger equations involving critical growth and discontinuous nonlinearities abstract in this paper we deal with a class of schrodinger equation in r n involving critical sobolev exponent and jump discontinuities the basic tool employed here is an approximation technique with periodic functions and variational arguments based on a linking theorem for locally lipschitz functionals

quasilinear asymptotically periodic elliptic equations with critical growth abstract the existence of a nontrivial solution for a class of asymptotically periodic equations involving the p laplacian operator in the euclidean space is established the solutions are obtained by an application of the mountain pass theorem and the concentration compactness principle the lack of compactness due to the fact that the domain is unbounded is compensated by comparing the minimax level associated with the original problem and the minimax level associated with the periodic limit problem

using fuzzy set theoretic techniques to identify preference rules from interactions in the linear model an empirical study abstract this paper seeks to establish a parametric linkage between fuzzy set theoretic techniques and commonly used preference formation rules in psychology and marketing such a linkage helps to benefit both fields we accomplish this objective by using a linear model with interaction term which nests many common preference protocols conjunction fuzzy and disjunction fuzzy or counterbalance fuzzy xor and linear compensatory the resulting linear model with interactions can be employed when one has no a priori hypothesis about the individual s preference formation rule involved to determine the most likely preference rule or to test more formally the adequacy of a given rule one illustrative application studies two attribute decisions in six product categories and demonstrates differences in preference formation processes by product category a second application demonstrates how fuzzy logical operators can be applied to situations involving more than two attributes

on nonlinear perturbations of a periodic elliptic problem in r2 involving critical growth we consider the equation u v x u f x u for x r2 where v r2 r is a positive potential bounded away from zero and the nonlinearity f r2 r r behaves like exp u 2 as u we also assume that the potential v x and the nonlinearity f x u are asymptotically periodic at infinity we prove the existence of at least one weak positive solution u h1 r2 by combining the mountain pass theorem with trudinger moser inequality and a version of a result due to lions for critical growth in r2

nonlinear perturbations of a periodic elliptic problem with critical growth abstract using the well known mountain pass theorem due to ambrosetti and rabinowitz we establish conditions for the existence of a positive solution for a class of nonlinear elliptic problems in r n with critical growth

polynomial inverse integrating factors for quadratic differential systems we characterize all the quadratic polynomial differential systems having a polynomial inverse integrating factor and provide explicit normal forms for such systems and for their associated first integrals we also prove that these families of quadratic systems have no limit cycles

a critical nonlinear fractional elliptic equation with saddle like potential in ℝn in this paper we study the existence of positive solution for the following class of fractional elliptic equation 2s su v z u uq 2u u2s 2uinrn where 0 are positive parameters q 2 2s 2s 2nn 2s n 2s s 0 1 su is the fractional laplacian and v is a saddle like potential the result is proved by using minimizing method constrained to the nehari manifold a special minimax level is obtained by using an argument made by benci and cerami

probabilistic subset conjunction the authors introduce subset conjunction as a classification rule by which an acceptable alternative must satisfy some minimum number of criteria the rule subsumes conjunctive and disjunctive decision strategies as special cases

schrodinger equations with asymptotically periodic terms we study the existence of non trivial solutions for a class of asymptotically periodic semilinear schrodinger equations in n by combining variational methods and the concentration compactness principle we obtain a non trivial solution for the asymptotically periodic case and a ground state solution for the periodic one in the proofs we apply the mountain pass theorem and its local version

ground states of degenerate quasilinear schrodinger equation with vanishing potentials abstract in this paper we study the existence of nontrivial ground state solutions for the following class of p laplacian type equation div a x u v x x p u p 2 u k x x p f u in r n where 1 p n n 3 n p p e 1 d 1 e p p e n p n d p critical hardy sobolev exponent f has a quasicritical growth v and k are nonnegative potentials the function a satisfies a x u c 0 x p h 0 x u p 1 c 0 1 x p h 1 x u p 1 for any r n a e x r n for any two positive functions h 1 l l o c r n h 0 l p p 1 r n with p p

liouvillian first integrals of differential equations liouvillian functions are functions that are built up from rational functions using exponentiation integration and algebraic functions we show that if a system of differential equations has a generic solution that satisfies a liouvillian relation that is there is a liouvillian function of several variables vanishing on the curve defined by this solution then the system has a liouvillian first integral that is a nonconstant liouvillian function that is constant on solution curves in some nonempty open set we can refine this result in special cases to show that the first integral must be of a very special form

an abstract approach to multibump solutions of periodic schrodinger equations and applications abstract we review a recent result on nonlinear superposition of zeros of asymptotically additive equivariant vector fields as application we present existence results for multibump solutions of nonlinear periodic schrodinger equations here 0 may be in a gap of the spectrum of the schrodinger operator and the nonlinearity may be of convolution type

positive solutions for some quasilinear equations with critical and supercritical growth abstract we establish results concerning the existence and multiplicity of positive solutions for the problem div a e x u p 2 u u p 1 f u u p 1 in r n u w 1 p r n where e 0 is a small parameter 2 p n p n p n p a is a positive potential and f is a superlinear function we obtain the existence of a ground state solution and relate the number of positive solutions with the topology of the set where a attains its minimum we also prove a multiplicity result for a supercritical version of the above problem in the proofs we use minimax theorems and ljusternik schnirelmann theory

nontrivial solutions for a class of semilinear biharmonic problems involving critical exponents in this paper we will discuss the existence of a nontrivial solution of the following class of semilinear biharmonic problem involving critical exponents 6u a x u h x u q 1u k x u p 1u in r u h r n 5 p where 1iqip 2 1 n 4 n 4 and a h k rn r are bounded nonnegative and continuous functions we recall that semilinear and quasilinear problems for second order equations have been extensively studied in the last years for laplace operator and subcritical case they have been studied for instance by rabinowitz 30 coti zelati and rabinowitz 14 kryszewski and szulkin 21 alama and li 1 when a h and k are 1 periodic while in the paper 25 montecchiari has dropped the periodicity condition on a h and

efficient algorithms for computing rational first integrals and darboux polynomials of planar polynomial vector fields we present fast algorithms for computing rational first integrals with bounded degree of a planar polynomial vector field our approach builds upon a method proposed by ferragut and giacomini whose main ingredients are the calculation of a power series solution of a first order differential equation and the reconstruction of a bivariate polynomial annihilating this power series we provide explicit bounds on the number of terms needed in the power series this enables us to transform their method into a certified algorithm computing rational first integrals via systems of linear equations we then significantly improve upon this first algorithm by building a probabilistic algorithm with arithmetic complexity o n 2 omega and a deterministic algorithm solving the problem in at most o d 2n 2 omega 1 arithmetic operations where n denotes the given bound for the degree of the rational first integral and where d leq n is the degree of the vector field and omega the exponent of linear algebra we also provide a fast heuristic variant which computes a rational first integral or fails in o n omega 2 arithmetic operations by comparison the best previous algorithm uses at least d omega 1 n 4 omega 4 arithmetic operations we then show how to apply a similar method to the computation of darboux polynomials the algorithms are implemented in a maple package rationalfirstintegrals which is available to interested readers with examples showing its efficiency

existence of solutions for a quasilinear schrodinger equation with vanishing potentials abstract in this paper we investigate the existence of positive solutions for the quasilinear schrodinger equation u v x u u 2 u g u in r n where n 3 g has a quasicritical growth and v is a nonnegative potential which can vanish at infinity

une generalisation de l inegalite de stein lorenzini soient k un corps de caracteristique quelconque k une cloture algebrique de k et p un polynome non constant de n 2 variables a coefficients dans k pour k on note n le nombre de facteurs irreductibles distincts f i d une decomposition de p sur k nous montrons l enonce suivant qui generalise des resultats de stein 1989 et lorenzini 1993 si le polynome p est non compose sur k alors n 1 est au plus egal a min ideg f i 1

c1 α local regularity of weak solutions of degenerate elliptic equations abstract it is demonstrated the local c 1 alpha nature of weak solutions of elliptic equations of the type 1 1 in the introduction under the degeneracy or singularity assumptions a sub 1 a sub 3

critical fractional p laplacian problems with possibly vanishing potentials abstract we obtain nontrivial solutions of a critical fractional p laplacian equation in the whole space and with possibly vanishing potentials in addition to the usual difficulty of the lack of compactness associated with problems involving critical sobolev exponents the problem is further complicated by the absence of a direct sum decomposition suitable for applying classical linking arguments we overcome this difficulty using a generalized linking construction based on the z 2 cohomological index

multiple solutions for nonhomogeneous elliptic equations involving critical sobolev exponent we consider the following problem where is continuous on r n and h x 0 by using ekeland s variational principle and the mountain pass theorem without ps conditions through a careful inspection of the energy balance for the approximated solutions we show that the probelm has at least two solutions for some 0 and 0 in particular if p 2 in a different way we prove that problem with 1 and h x 0 has at least two positive solutions as

bounding the number of remarkable values via jouanolou s theorem abstract in this article we bound the number of remarkable values of a polynomial vector field the proof is short and based on jouanolou s theorem about rational first integrals of planar polynomial derivations our bound is given in term of the size of a newton polygon associated to the vector field we prove that this bound is almost reached

existence of solutions for a class of nonlinear schrodinger equations with potential vanishing at infinity abstract in this paper we investigate the existence of positive ground state solution for the following class of elliptic equations u v x u k x f u in r n where n 3 v k are nonnegative continuous functions and f is a continuous function with a quasicritical growth here we prove a hardy type inequality and use it together with variational method to get a ground state solution

darboux theory of integrability in the sparse case darboux s theorem and jouanolou s theorem deal with the existence of first integrals and rational first integrals of a polynomial vector field these results are given in terms of the degree of the polynomial vector field here we show that we can get the same kind of results if we consider the size of a newton polytope associated to the vector field furthermore we show that in this context the bound is optimal

non autonomous perturbations for a class of quasilinear elliptic equations on r abstract this paper is concerned with the existence of two positive solutions for a class of quasilinear elliptic equations on r involving the p laplacian with a non autonomous perturbation the first solution is obtained as a local minimum in a neighborhood of 0 and the second one by a mountain pass argument the special features of the problem here is the complex structure of the linear part which in particular oblige to work into the space w 1 p r then one faces problems in the convergence of the sequences of derivatives u n u

solutions in spectral gaps for a nonlinear equation of schrodinger type abstract in this paper we study the existence of a nontrivial h2 r n solution for an equation of the form u x p x u x f x u x u x x r n r where p l r n is a periodic function we assume that the operator p h2 r n l2 r n l2 r n is strongly indefinite and invertible and that x r r is odd and satisfies some superlinear but subcritical growth conditions we extend the class of nonlinearities which has been studied up to now in particular under standard technical restrictions the existence of a solution is derived when lim x f x s f s 0 exists for all s r if we assume that f x s f s for all s r and a e on r n

on a periodic schrodinger equation involving periodic and nonperiodic nonlinearities in r2 abstract we study the existence of solutions for the nonlinear schrodinger equation u v x u f x u in r 2 where the potential v is 1 periodic 0 lies in a spectral gap from the spectrum of the schrodinger operator s v and the nonlinearity f x t has exponential growth in the sense of trudinger moser the main feature here is that f x t is allowed to be both periodic and nonperiodic in the x variable our proofs rely on a linking theorem and the lions concentration compactness principle

on a periodic schrodinger equation with nonlocal superlinear part we consider the choquard pekar equation open image in new window and focus on the case of periodic potential v for a large class of even functions w we show existence and multiplicity of solutions essentially the conditions are that 0 is not in the spectrum of the linear part v and that w does not change sign our results carry over to more general nonlinear terms in arbitrary space dimension n 2

estimating multiple classification latent class models this paper presents a new class of models for persons by items data the essential new feature of this class is the representation of the persons every person is represented by its membership tomultiple latent classes each of which belongs to onelatent classification the models can be considered as a formalization of the hypothesis that the responses come about in a process that involves the application of a number ofmental operations two algorithms for maximum likelihood ml and maximum a posteriori map estimation are described they both make use of the tractability of the complete data likelihood to maximize the observed data likelihood properties of the map estimators i e uniqueness and goodness of recovery and the existence of asymptotic standard errors were examined in a simulation study then one of these models is applied to the responses to a set of fraction addition problems finally the models are compared to some related models in the literature

existence and bifurcation of solutions for nonlinear perturbations of the periodic schrodinger equation abstract in this paper we apply previously obtained abstract bifurcation results to nonlinear perturbations of the periodic schrodinger equation we show that depending on the sign of the nonlinearity lower or upper end points of the continuous spectrum are bifurcation points the main part of the proof consists of the construction of suitable test functions which are required for the application of the abstract theorem instead of pursuing the notion of an eigenpaket of the differential operator which has successfully been used in the one dimensional case we now exploit a construction based on bloch waves of the linear schrodinger operator

remarks on sublinear elliptic problems on unbounded cylinders m l where v s o r o r is a bounded smooth domain m l g 0 m w n m q l g 1 v s o if m s 0 and v s r if l s 0 f v 0 a r is a caratheodory function and r v a r is continuous r g 0 there is by now an extensive literature on this kind of problem for w x w x w x w x instance amick and toland 1 stuart 2 esteban 3 burton 4 bona w x w x bose and turner 5 and berestycki and lions 20 where different conditions on r x and f x s as well as several techniques were considered this list is of course far from complete and we would also refer to w x w x schindler 6 costa and miyagaki 7 and their references physical motiva

existence of solutions for semilinear elliptic equations with indefinite linear part we study equations on rn lu w x u p 1 ulu f u where l is an indefinite schrodinger operator 0 w x 0 as and subcritical nonlinearities we prove existence of solutions u h1 rn via an inversion to a dual variational problem

existence of a ground state solution for an elliptic problem with critical growth in an exterior domain abstract in this work we prove the existence of a radially symmetric ground state solution for an elliptic equation in an exterior of a ball with neumann boundary condition

existence of solution for two classes of elliptic problems in r n with zero mass in this paper we show the existence of a positive solution for the following class of elliptic equations u k x f u in rn where f is a continuous function with quasicritical growth and k is a nonnegative function verifying some conditions of two kinds we analyze the problem with k asymptotically periodic k periodic and k belonging to some lr

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

mountain pass solutions to equations of p laplacian type abstract this work is devoted to study the existence of solutions to equations of p laplacian type we prove the existence of at least one solution and under further assumptions the existence of infinitely many solutions in order to apply mountain pass results we introduce a notion of uniformly convex functional that generalizes the notion of uniformly convex norm

on the total order of reducibility of a pencil of algebraic plane curves in this paper the problem of bounding the number of reducible curves in a pencil of algebraic plane curves is addressed unlike most of the previous related works each reducible curve of the pencil is here counted with its appropriate multiplicity it is proved that this number of reducible curves counted with multiplicity is bounded by d 2 1 where d is the degree of the pencil then a sharper bound is given by taking into account the newton s polygon of the pencil

a nonhomogeneous elliptic problem involving critical growth in dimension two abstract in this paper we study a class of nonhomogeneous schrodinger equations u v x u f u h x in the whole two dimension space where v x is a continuous positive potential bounded away from zero and which can be large at the infinity the main difficulty in this paper is the lack of compactness due to the unboundedness of the domain besides the fact that the nonlinear term f s is allowed to enjoy the critical exponential growth by means of the trudinger moser inequality by combining variational arguments and a version of the trudinger moser inequality we establish the existence of two distinct solutions when h is suitably small

local behavior of planar analytic vector fields via integrability we present an algorithm to study the local behavior of singular points of planar analytic vector fields having a first integral which is a quotient of analytic functions the algorithm is based on the blow up method it emphasizes the curves passing through the singular points and avoids the computation of the desingularized systems vector fields having a rational first integral are a particular case

nonlinear perturbations of a periodic kirchhoff equation in rn abstract this paper is concerned with the existence of positive solutions for the following class of nonlocal problem m r n u 2 d x r n v x u 2 d x u v x u f u u in r n where 5 for n 3 and 1 for n 1 2 is a positive parameter and 0 1 moreover m v and f are continuous functions satisfying some conditions

on the remarkable values of the rational first integrals of polynomial vector fields abstract the remarkable values for polynomial vector fields in the plane having a rational first integral were introduced by poincare he was mainly interested in their algebraic aspects here we are interested in their dynamic aspects i e how they contribute to the phase portrait of the system to its separatrices to its singular points etc the relationship between remarkable values and dynamics mainly takes place through the inverse integrating factor

existence of solutions for a class of elliptic equations in r n with vanishing potentials in this paper we investigate the existence of positive solution for the following class of elliptic equation u v x u f u in rn where f has a subcritical growth and v is a nonnegative potential which can vanish at infinity that is v x 0 as x or shortly v 0

ground state solution for a class of indefinite variational problems with critical growth abstract in this paper we study the existence of ground state solution for an indefinite variational problem of the type p u v x w x u f x u in r n u h 1 r n where n 2 v w r n r and f r n r r are continuous functions verifying some technical conditions and f possesses a critical growth here we will consider the case where the problem is asymptotically periodic that is v is z n periodic w goes to 0 at infinity and f is asymptotically periodic

soliton solutions for quasilinear schrodinger equations with critical growth in this paper we establish the existence of standing wave solutions for quasilinear schrodinger equations involving critical growth by using a change of variables the quasilinear equations are reduced to semilinear one whose associated functionals are well defined in the usual sobolev space and satisfy the geometric conditions of the mountain pass theorem using this fact we obtain a cerami sequence converging weakly to a solution v in the proof that v is nontrivial the main tool is the concentration compactness principle due to p l lions together with some classical arguments used by h brezis and l nirenberg 1983 in 9

multiple solutions for p laplacian type equations abstract in this paper we establish the existence of three weak solutions of an equation which involves a general elliptic operator in divergence form in particular a p laplacian operator while the nonlinearity has a p 1 sublinear growth at infinity this result completes some recent papers where mountain pass type solutions were obtained providing the nonlinear term via a p 1 superlinear growth at infinity fulfilling an ambrosetti rabinowitz type condition in our case an abstract critical point result is applied proved by g bonanno g bonanno some remarks on a three critical points theorem nonlinear analysis 54 2003 651 665

infinitely many stationary solutions of discrete vector nonlinear schrodinger equation with symmetry in this paper we study the existence of stationary solutions for the following discrete vector nonlinear schrodinger email protected email protected email protected email protected f email protected email protected n jf n f n f n where f n is a sequence of 2 component vector j 0110 email protected email protected n email protected n email protected nis the discrete laplacian in one spatial dimension and sequence t n is assumed to be n periodic in n i e t n email protected n we prove the existence of infinitely many nontrivial stationary solutions for this system by variational methods the same method can also be applied to obtain infinitely many breather solutions for single discrete nonlinear schrodinger equation