correlation functions in the itzykson zuber model then point function for the integral over unitary matrices with itzykson zuber measure is reduced to the integral over the gelfand tzetlin table the integrand for genericn is given by linear exponential times the rational function forn 2 and in some cases forn 2 later in fast is the polynomial and this allows to give an explicit and simple expression for all 2 point and a set ofn point functions for the most generaln point function a simple linear differential equation is constructed

harnack estimates and extinction profile for weak solutions of certain singular parabolic equations we establish an intrinsic harnack estimate for nonnegative weak solutions of the singular equation 1 1 below for m in the optimal range n 2 n 1 intrinsic means that due to the singularity the space time dimensions in the parabolic geometry must be rescaled by a factor determined by the solution itself consequences are sharp supestimates on the solutions and decay rates as t approaches the extinction time analogous results are shown for p laplacian type equations

an analysis of iriscode iriscode is an iris recognition algorithm developed in 1993 and continuously improved by daugman it has been extensively applied in commercial iris recognition systems iriscode representing an iris based on coarse phase has a number of properties including rapid matching binomial impostor distribution and a predictable false acceptance rate because of its successful applications and these properties many similar coding methods have been developed for iris and palmprint identification however we lack a detailed analysis of iriscode the aim of this paper is to provide such an analysis as a way of better understanding iriscode extending the coarse phase representation to a precise phase representation and uncovering the relationship between iriscode and other coding methods our analysis demonstrates that iriscode is a clustering algorithm with four prototypes the locus of a gabor function is a 2 d ellipse with respect to a phase parameter and can be approximated by a circle in many cases gabor function can be considered as a phase steerable filter and the bitwise hamming distance can be regarded as a bitwise phase distance we also discuss the theoretical foundation of the impostor binomial distribution we use this analysis to develop a precise phase representation which can enhance accuracy finally we relate iriscode and other coding methods

character expansion method for supergroups and extended superversions of the leutwyler smilga and berezin karpelevich integrals we introduce an extension of the character expansion method to the case of supergroups this method allows us to calculate a superversion of the leutwyler smilga integral which to the best of our knowledge has not been calculated before we also use the method to generalize a previously calculated superversion of the berezin karpelevich integral our character expansion method should also allow for the calculation of other supergroup integrals

on some integrals over the u n unitary group and their large n limit the integral over the u n unitary group i du exp tr au bu is re examined various approaches and extensions are first reviewed the second half of the paper deals with more recent developments relation with integrable toda lattice hierarchy diagrammatic expansion and combinatorics and what they teach us on the large n limit of log i

higher holder regularity for the fractional p laplacian in the superquadratic case abstract we prove higher holder regularity for solutions of equations involving the fractional p laplacian of order s when p 2 and 0 s 1 in particular we provide an explicit holder exponent for solutions of the non homogeneous equation with data in l q and q n s p which is almost sharp whenever s p p 1 n q the result is new already for the homogeneous equation

optimal wloc2 2 regularity pohozhaevʼs identity and nonexistence of weak solutions to some quasilinear elliptic equations abstract we begin by establishing a sharp optimal w loc 2 2 regularity result for bounded weak solutions to a nonlinear elliptic equation with the p laplacian p u def div u p 2 u 1 p we develop very precise optimal regularity estimates on the ellipticity of this degenerate for 2 p or singular for 1 p 2 problem we apply this regularity result to prove pohozhaev s identity for a weak solution u w 1 p of the elliptic neumann problem p p u w u f x in u 0 on here is a bounded domain in r n whose boundary is a c 2 manifold x 0 denotes the outer unit normal to at x 0 x x 1 x n is a generic point in and f l w 1 1 the potential w r r is assumed to be of class c 1 and of the typical double well shape of type w s 1 s for s r where 1 are some constants finally we take an advantage of the pohozhaev identity to show that problem p with f 0 in has no phase transition solution u w 1 p 1 p n such that 1 u 1 in with u 1 in 1 and u 1 in 1 where both 1 and 1 are some nonempty subdomains of such a scenario for u is possible only if n 1 and 1 1 are finite unions of suitable subintervals of the open interval r 1

extremal solutions of p laplacian problems in d1 p rn via wolff potential estimates abstract we prove the existence of smallest positive and greatest negative solutions in d 1 p r n of the p laplacian problem of logistic type in r n with 1 p n u d 1 p r n p u a x u p 2 u g u and consider its extended version with a harvesting term h as well unlike papers on the topic here we only assume a superlinear growth for the nonlinearity g r r at zero and at infinity i e even supercritical growth is allowed and moreover no monotonicity for s g s s p 1 is required we prove a vazquez type maximum principle for nonnegative d 1 p r n solutions of some p laplacian problem and establish a sub supersolution method in d 1 p r n which along with wolff potential estimates for superharmonic functions will be used as our main tools

existence and multiplicity of positive solutions for semilinear elliptic systems with sobolev critical exponents abstract in this paper we study the existence and multiplicity of positive solutions to the following system u f u u v e g x v f v u v e h x in u v 0 in and u v 0 on where is a bounded smooth domain in r n f c 1 r 2 r is positively homogeneous of degree g h c 1 0 and e is a positive parameter using sub supersolution method we prove the existence of positive solutions for the above problem by means of the variational approach we prove the multiplicity of positive solutions for the above problem with 2 2

angular gelfand tzetlin coordinates for the supergroup uosp k 1 2k 2 we construct gelfand tzetlin coordinates for the unitary orthosymplectic supergroup uosp k1 2k2 this extends a previous construction for the unitary supergroup u k1 k2 we focus on the angular gelfand tzetlin coordinates i e our coordinates stay in the space of the supergroup we also present a generalized gelfand pattern for the supergroup uosp k1 2k2 and discuss various implications for representation theory

improved iris recognition through fusion of hamming distance and fragile bit distance the most common iris biometric algorithm represents the texture of an iris using a binary iris code not all bits in an iris code are equally consistent a bit is deemed fragile if its value changes across iris codes created from different images of the same iris previous research has shown that iris recognition performance can be improved by masking these fragile bits rather than ignoring fragile bits completely we consider what beneficial information can be obtained from the fragile bits we find that the locations of fragile bits tend to be consistent across different iris codes of the same eye we present a metric called the fragile bit distance which quantitatively measures the coincidence of the fragile bit patterns in two iris codes we find that score fusion of fragile bit distance and hamming distance works better for recognition than hamming distance alone to our knowledge this is the first and only work to use the coincidence of fragile bit locations to improve the accuracy of matches

integration over matrix spaces with unique invariant measures we present a method to calculate integrals over monomials of matrix elements with invariant measures in terms of wick contractions the method gives exact results for monomials of low order for higher order monomials it leads to an error of order 1 n where n is the dimension of the matrix and where is independent of the degree of the monomial we give a lower bound on the integer and show how can be increased systematically the method is particularly suited for symbolic computer calculation explicit results are given for o n u n and for the circular orthogonal ensemble

gelfand tzetlin coordinates for the unitary supergroup the gelfand tzetlin method provides explicit coordinates on the parameter space of the unitary groupu k which make direct evaluations of group integrals possible it is closely related to the gelfand construction of finite dimensional irreducible representations we generalize the gelfand tzetlin method to the unitary supergroupu k1 k2 the coordinates on the parameter space for supergroup integrals and the invariant haar measure are evaluated as an example the supersymmetric harish chandra itzykson zuber integral is calculated a generalized gelfand pattern containing anticommuting variables is introduced which determines the representation

existence of a non trivial solution for the p laplacian equation with fucik type resonance at infinity abstract we prove the existence of a non trivial weak solution for the equation p u a u p 1 b u p 1 f x u in u 0 on by variational methods we consider the case where a 1 or b 1 1 denotes the first eigenvalue of p or the case where a is close to b while the nonlinearity f belongs to a class that contains for example u q 2 u 1 q p

existence and nonexistence of nontrivial solutions of some nonlinear degenerate elliptic equations in connection with the maximizing problem for the functional r u u lq u lp in w01 p 0 we consider the equation div u p 2 u x u q 2 u x x 1 p resp q p e has no nontrivial resp nonnegative nontrivial solution belonging to the class p u w01 p lq xi u q 2u lp p 1 i 1 2 n w01 p provided that is star shaped the crucial point of the proof of our result is to obtain an l estimate of weak solutions and to verify a certain pohozaev type inequality for weak solutions belonging to p

recursive construction for a class of radial functions ii superspace we extend the recursion formula for matrix bessel functions which we obtained previously to superspace it is sufficient to do this for the unitary orthosymplectic supergroup by direct computations we show that fairly explicit results can be obtained at least up to dimension 8 8 for the supermatrices since we introduce a new technique we discuss various of its aspects in some detail

global l gradient estimates for solutions to a certain degenerate elliptic equation abstract in view of the applications in the study of regularity properties of minimizers for a continuous model of transportation which is a kind of divergence constrained optimization problem we prove a global l gradient estimate for solutions of an elliptic equation whose ellipticity constants become degenerate at every point where u with 0 the exposition is as self contained as possible

the design and use of steerable filters the authors present an efficient architecture to synthesize filters of arbitrary orientations from linear combinations of basis filters allowing one to adaptively steer a filter to any orientation and to determine analytically the filter output as a function of orientation steerable filters may be designed in quadrature pairs to allow adaptive control over phase as well as orientation the authors show how to design and steer the filters and present examples of their use in the analysis of orientation and phase angularly adaptive filtering edge detection and shape from shading one can also build a self similar steerable pyramid representation the same concepts can be generalized to the design of 3 d steerable filters

derivation of the supersymmetric harish chandra integral for uosp k 1 2k 2 the previous supersymmetric generalization of the unitary harish chandra integral prompted the conjecture that the harish chandra formula should have an extension to superspaces we prove this conjecture for the unitary orthosymplectic supermanifold uosp k1 2k2 to this end we construct and solve an eigenvalue equation

spectral theory for linearized p laplace equations abstract we continue and completely set up the spectral theory initiated in castorina et al 2009 5 for the linearized operator arising from p u f u 0 we establish existence and variational characterization of all the eigenvalues and by a weak harnack inequality we deduce holder continuity for the corresponding eigenfunctions this regularity being sharp the morse index of a positive solution can be now defined in the classical way and we will illustrate some qualitative consequences one should expect to deduce from such information in particular we show that zero morse index or more generally non degenerate solutions on the annulus are radial

generalized minimizer solutions for equations with the p laplacian and a potential term let varomega be a domain in mathbb r n possibly unbounded n geq2 1 and let v in l mathrm loc infty varomega consider the energy functional mathcal q v on c mathrm c infty varomega and its g teaux derivative mathcal q v prime respectively given by mathcal q v u eqdef frac 1 p int varomega nabla u p v u p mathrm d x qquad mathcal q v prime u mathrm div nabla u p 2 nabla u v u p 2 u for u in c mathrm c infty varomega assume that mathcal q v 0 on c mathrm c infty varomega setminus 0 and that mathcal q v does not have a ground state in the sense of a null sequence for mathcal q v that converges in l mathrm loc p varomega to a positive function varphi in c mathrm loc 1 varomega a ground state finally let f in mathcal d prime varomega be such that the functional u mapsto mathcal q v u langle u f rangle c mathrm c infty varomega to mathbb r is bounded from below then the equation mathcal q v prime u f has a solution u 0 in w 1 p mathrm loc varomega in the sense of distributions this solution also minimizes the functional u mapsto mathcal q v u langle u f rangle c mathrm c infty varomega to mathbb r where mathcal q v denotes the bipolar vargamma regularization of mathcal q v and mathcal q v is the largest convex weakly lower semicontinuous functional on c mathrm c infty varomega that satisfies mathcal q v leq mathcal q v the original energy functional mathcal q v is not necessarily convex

uniqueness of positive radial solutions for quasilinear elliptic equations in an annulus abstract extending a previous result of tang 1 we prove the uniqueness of positive radial solutions of p u f u 0 subject to dirichlet boundary conditions on an annulus in r n with 2 p n under suitable hypotheses on the nonlinearity f this argument also provides an alternative proof for the uniqueness of positive solutions of the same problem in a finite ball see 9 in the complement of a ball or in the whole space r n see 10 3 11

recursive construction for a class of radial functions i ordinary space a class of spherical functions is studied which can be viewed as the matrix generalization of bessel functions we derive a recursive structure for these functions we show that they are only special cases of more general radial functions which also have a properly generalized recursive structure some explicit results are worked out for the first time we identify a subclass of such radial functions which consist of a finite number of terms only

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

bounded perturbations of homogeneous quasilinear operators using bifurcations from infinity this paper deals with existence results for the following nonlinear problem with the dirichlet p laplacian p in a bounded domain rn pu u p 2u h x u in u 0on here pu defdiv u p 2 u where p 1 is a fixed number h h x s is a given function from r into r and r stands for a spectral parameter we focus on close to 1 including the resonant case 1 the nonlinearity h is assumed to be of landesman lazer type but we can deal with vanishing nonlinearities as well our asymptotic method substitutes the lyapunov schmidt method in some sense unlike in the semilinear case p 2 our method can treat more general nonlinearities if p 2 vanishing nonlinearities with very fast decay

regularity and existence of solutions of elliptic equations with p q growth conditions for some positive constants m a4 and for exponents q 2 p 2 2 under 1 2 1 3 and some other assumptions by assuming also that the quotient q p is sufficiently close to one in dependence on n precisely if q p c n n 2 then we prove that every weak solution to 1 1 of class 4 4 0 is locally lipschitz continuous in q moreover there are positive constants 8 c and 0 2 1 such that

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

the existence and non existence of positive solutions to a singular quasilinear elliptic problem in rn abstract this paper deals with the existence and nonexistence of entire positive solutions of the quasilinear elliptic equation p u a x g u b x f u in r n 1 p n n 3 with u x 0 as x here either g or f or both of them are singular at 0 in the sense that g t f t as t 0 by using a perturbation method which eliminates the singularity on a ball with radius r and then let r tends to infinity with help of the bounded super solution of the original problem we obtain the existence of a weak solution of the problem the main results of this paper improve the corresponding results of dragos patru covei the existence and asymptotic behavior of a positive solution to a quasilinear elliptic problem in r n nonlinear anal 69 2008 2615 2622 j v goncalves c a santos singular elliptic problems existence non existence and boundary behavior nonlinear anal 66 2007 2078 2090

global lorentz estimates for nonuniformly nonlinear elliptic equations via fractional maximal operators abstract this article is a contribution to the study of regularity theory for nonlinear elliptic equations the aim of this article is to establish some global estimates for nonuniformly elliptic equations in divergence form as div u p 2 u a x u q 2 u div f p 2 f a x f q 2 f which arises from double phase functional problems in particular the main results provide the regularity estimates for the distributional solutions in terms of maximal and fractional maximal operators this work extends that of colombo and mingione 2016 23 and byun and oh 2017 9 by dealing with the global estimates in lorentz spaces it also extends our recent result tran and nguyen 2020 66 concerning new estimates of divergence elliptic equations using cutoff fractional maximal operators for future research the approach developed in this article will allow global estimates of distributional solutions to nonuniformly nonlinear elliptic equations to be obtained in the framework of other spaces

on fractional regularity methods for a class of nonlocal problems abstract in the past years the phenomenon of fractional regularity has been addressed for a large class of linear and or quasilinear differential operators mostly in terms of certain besov spaces as it turned out for equations governed by the p laplacian in general the regularity of solutions appears in terms of functional spaces with nonlinear order of smoothness moreover despite its own interest fractional regularity methods may be used as a tool for the investigation of some partial differential equations which are not usually addressed in this manner thus the purpose of the present paper is to exploit such methods in order to provide some results regarding existence and regularity of solutions to a class nonlocal elliptic equations which are linked to the p laplacian this is done by means of explicit a priori estimates regarding lebesgue and nikolskii spaces which are part of the present contribution as a consequence this approach allows a relaxation on some of the standard conditions employed in this class of problems

2 matrix versus complex matrix model integrals over the unitary group as triangular integrals we prove that the 2 hermitian matrix model and the complex matrix model obey the same loop equations and as a byproduct we find a formula for itzykzon zuber type integrals over the unitary group integrals over u n are rewritten as gaussian integrals over triangular matrices and then computed explicitly that formula is an efficient alternative to the former shatashvili s formula

kazakov migdal model with logarithmic potential and the double penner matrix model the kazakov migdal km model is a u n lattice gauge theory with a scalar field in the adjoint representation but with no kinetic term for the gauge field this model is formally soluble in the limit n though explicit solutions are available for a very limited number of scalar potentials a double penner model in which the potential has two logarithmic singularities provides an example of an explicitly soluble model the formal solution to this double penner km model is reviewed first special attention is paid to the relationship of this model to an ordinary one matrix model whose potential has two logarithmic singularities the double penner model a detailed analysis is presented of the large n behavior of this double penner model the various one cut and two cut solutions are described and cases in which eigenvalue condensation occurs at the singular points of the potential are discussed then the consequences of our study for the km model described above are discussed the phase diagram

existence uniqueness and qualitative properties of the solution of a degenerate nonlinear parabolic system abstract in this paper we are concerned with the study of the existence of solutions of a nonlinear parabolic system with density dependent diffusion subject to homogeneous third type boundary conditions a comparison result is shown that allows the study of the asymptotic behavior of the solutions via the same methods used for systems with classical diffusion finally an application of the results to an epidemic system is presented sufficient conditions are given for the positivity of the solution of this system

some properties of angular integrals we find new representations for itzykson zuber like angular integrals for arbitrary in particular for the orthogonal group o n the unitary group u n and the symplectic group sp 2n we rewrite the haar measure integral as a flat lebesge measure integral and we deduce some recursion formulae on n the same methods give also shatashvili s type moments finally we prove that in agreement with brezin and hikami s observation the angular integrals are linear combinations of exponentials whose coefficients are polynomials in the reduced variables xi xj yi yj

two different approaches for iris recognition using gabor filters and multiscale zero crossing representation abstract importance of biometric user identification is increasing everyday one of the most promising techniques is the one based on the human iris the authors in this work describe different approaches to develop this biometric technique based on the works carried out by daugman the authors have worked using gabor filters and hamming distance but in addition they have also worked in zero crossing representation of the dyadic wavelet transform applied to two different iris signatures one based on a single virtual circle of the iris the other one based on an annular region also other metrics have been applied to be compared with the results obtained with the hamming distance in this work euclidean distance and d z will be shown the last proposed approach is translation rotation and scale invariant results will show a classification success up to 99 6 achieving an equal error rate down to 0 12 and the possibility of having null false acceptance rates with very low false rejection rates

regularity for quasilinear equations on degenerate singular sets we prove a new universal gradient continuity estimate for solutions to quasilinear equations with varying coefficients at singular set of degeneracy fancyscript s u x d u x 0 our main theorem reveals that along fancyscript s u u is asymptotically as regular as solutions to constant coefficient equations in particular along the critical set fancyscript s u du enjoys a modulus of continuity much superior than the possibly low continuity feature of the coefficients the results are new even in the context of linear elliptic equations where it is herein shown that h 1 weak solutions to text div left a ij x du right 0 with a ij elliptic and dini continuous are actually c 1 1 along fancyscript s u the results and insights of this work foster a new understanding on smoothness properties of solutions to degenerate or singular equations beyond typical elliptic regularity estimates precisely where the diffusion attributes of the equation collapse

on multiplicity and concentration of positive solutions for a class of quasilinear problems with critical exponential growth in rn abstract in this work we study the existence multiplicity and concentration of positive solutions for the following class of quasilinear problems n n u v x u n 2 u f u in r n u x 0 x r n where n u is the n laplacian operator is a positive parameter n 2 v r n r is a continuous function and f r r is a c 1 function having critical exponential growth

travelling waves in a convection diffusion equation abstract we study existence and stability of travelling waves for nonlinear convection diffusion equations in the 1 d euclidean space the diffusion coefficient depends on the gradient in analogy with the p laplacian and may be degenerate unconditional stability is established with respect to initial data perturbations in l 1 r

rotational invariant estimator for general noisy matrices we investigate the problem of estimating a given real symmetric signal matrix c from a noisy observation matrix m in the limit of large dimension we consider the case where the noisy measurement m comes either from an arbitrary additive or multiplicative rotational invariant perturbation we establish using the replica method the asymptotic global law estimate for three general classes of noisy matrices significantly extending previously obtained results we give exact results concerning the asymptotic deviations called overlaps of the perturbed eigenvectors away from the true ones and we explain how to use these overlaps to clean the noisy eigenvalues of m we provide some numerical checks for the different estimators proposed in this paper and we also make the connection with some well known results of bayesian statistics

linear potentials in nonlinear potential theory pointwise gradient bounds via riesz potentials such as those available for the linear poisson equation actually hold for general quasilinear degenerate equations of p laplacean type the regularity theory of such equations completely reduces to that of the classical poisson equation up to the c1 level

on singular quasi monotone p q laplacian systems we combine the sub and supersolution method and perturbation arguments to obtain positive solutions of singular quasi monotone p q laplacian systems

correlation functions of harish chandra integrals over the orthogonal and the symplectic groups the harish chandra correlation functions i e integrals over compact groups of invariant monomials prod operatorname tr x p 1 varomega y q 1 varomega dagger x p 2 cdots with the weight exp tr x y are computed for the orthogonal and symplectic groups we proceed in two steps first the integral over the compact group is recast into a gaussian integral over strictly upper triangular complex matrices with some additional symmetries supplemented by a summation over the weyl group this result follows from the study of loop equations in an associated two matrix integral and may be viewed as the adequate version of duistermaat heckman s theorem for our correlation function integrals secondly the gaussian integration over triangular matrices is carried out and leads to compact determinantal expressions

on the gibbons conjecture for equations involving the p laplacian in this paper we prove the validity of gibbons conjecture for the quasilinear elliptic equation delta p u f u on mathbb r n the result holds true for 2n 2 n 2 p 2 and for a very general class of nonlinearity f

maximum principle of optimal control for degenerate quasi linear elliptic equations optimal control problems governed by degenerate quasi linear partial differential equations of elliptic type are considered the optimal control systems considered may lack cesari type conditions and therefore the corresponding approximate optimal control problem may have no solution to yield the maximum principle of optimal pairs relaxed controls are used to overcome the difficulties occurring when considering approximate problems the relaxed controls used are defined by finite additive measures so that the case of the control set being noncompact can be treated

up to boundary regularity for a singular perturbation problem of p laplacian type abstract in this paper we are interested in establishing up to boundary uniform estimates for the one phase singular perturbation problem involving a nonlinear singular degenerate elliptic operator our main result states if r n is a c 1 domain f c 1 for some 0 1 and u e verifies div a x u e u e e u e in 0 u e 1 in u e f on where e 0 e t 1 e t e and 0 t b 0 t 1 r e t d t m 0 with some positive constants b and m then there exists a constant c 0 independent of e such that u e l c

mixed correlation functions of the two matrix model we compute the correlation functions mixing the powers of two non commuting random matrices within the same trace the angular part of the integration was partially known in the literature morozov a 1992 mod phys lett a 7 3503 shatashvili s l 1993 commun math phys 154 421 we pursue the calculation and carry out the eigenvalue integration reducing the problem to the construction of the associated bi orthogonal polynomials the generating function of these correlations then becomes a determinant involving the recursion coefficients of the bi orthogonal polynomials

some remarks on the fucik spectrum of the p laplacian and critical groups let be a bounded domain in n n 1 for 1 p the fucik spectrum of the p laplacian on w 1 p 0 is defined as the set p of those points a b 2 such that pu a u p 1 b u p 1 u w 1 p 0 1 1 has a nontrivial solution here pu div u p 2 u and u max u 0 it is known that the first eigenvalue 1 of p on w 1 p 0 is positive simple and admits a positive eigenfunction 1 w 1 p 0 c1 see lindqvist 8 so p contains the two lines 1 and 1 a first nontrivial curve c2 in p through 2 2 where 2 is the second eigenvalue of p was recently constructed and variationally characterized by a

quasilinear elliptic problems under strong resonance conditions abstract in this paper we establish the existence and multiplicity of solutions for a quasilinear elliptic problem under strong resonance conditions at infinity in order to control the resonance we consider a new hypothesis on the nonlinear term in all results we use variational methods critical groups and the morse theory