the beginning of the fucik spectrum for the p laplacian has a nontrivial solution u here 1 p 2p u div u p 2 u u max u 0 u uu in 1 1 denoting by 1 2 the first two eigenvalues of 2p on w 1 p 0 0 cf the end of this introduction 7p clearly contains in particular 1 1 2 2 and the two lines 1 r and r 1 this generalized notion of spectrum was introduced for p 2 i e in the linear case in the 1970s by fuc ik 21 and dancer 15 in connection article id jdeq 1999 3645 available online at http www idealibrary com on

multiple solutions for p laplacian type problems with asymptotically p linear terms via a cohomological index theory abstract the aim of this paper is investigating the existence of weak solutions of the quasilinear elliptic model problem div a x u u p 2 u 1 p a t x u u p f x u in u 0 on where r n is a bounded domain n 2 p 1 a is a given function which admits partial derivative a t x t a t x t and f is asymptotically p linear at infinity under suitable hypotheses both at the origin and at infinity and if a x is even while f x is odd by using variational tools a cohomological index theory and a related pseudo index argument we prove a multiplicity result if p n in the non resonant case

nontrivial solutions for a class of quasilinear problems with jumping nonlinearities via a cohomological local splitting abstract we establish the existence of a nontrivial solution for a class of quasilinear elliptic boundary value problems with jumping nonlinearities by means of variational arguments and a cohomological local splitting

multiplicity of solutions for a fourth order quasilinear nonhomogeneous equation we consider a fourth order quasilinear nonhomogeneous equation which is equivalent to a nonhomogeneous hamiltonian system the purpose of this work is to prove the existence of at least two solutions for such equation when a certain parameter is small enough furthermore under an additional hypothesis on positiveness of the nonhomogeneous part we prove that our solutions are positive

sign changing and nontrivial solutions for a class of kirchhoff type problems abstract in this paper we study the existence of sign changing nodal and nontrivial solutions for the nonlinear kirchhoff type equation a b u 2 d x u u u 3 in u 0 on where r are two real parameters with the help of nodal nehari set we first provide a description of a two dimensional set in the plane which corresponds to the nonexistence and existence of sign changing solutions for the above kirchhoff type equation and then we establish the existence result of nontrivial solutions via the minimax methods

sign changing solutions for some fourth order nonlinear elliptic problems in this paper we consider the existence and multiplicity of sign changing solutions for some fourth order nonlinear elliptic problems and some existence and multiple are obtained the weak solutions are sought by means of sign changing critical theorems

variational identities and applications to differential systems j serrin 1 introduction in the work of ponos p and in recent work of pucci serrin ps variational identities are discussed which are useful for solving various ques tions about elliptic differential equations and particular elliptic systems here we obtain a number of further applications of these identities to systems it was noted in ps that these identities are closely related to noether s theorem for variational problems by working out the divergence term arising in noether s theorem and making some simplifications one obtains a general identity for variational problems if a variational integral has a symmetry group g with the following infinitesimal generator

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 of infinitely many weak solutions for the p laplacian with nonlinear boundary conditions abstract in this paper we deal with the existence of weak solutions for quasilinear elliptic problem involving a p laplacian of the form p u x u p 2 u f x u in u p 2 u u p 2 u on we consider the above problem under several conditions on f for f superlinear and subcritical with respect to u we prove the existence of infinitely many solutions of the above problem by using the fountain theorem and the dual fountain theorem respectively for the case where f is critical with a subcritical perturbation namely f x u u p 2 u u r 2 u we show that there exists at least a nontrivial solution when p r p and there exist infinitely many solutions when 1 r p by using the mountain pass theorem and the concentration compactness principle respectively

multiple solutions for asymptotically p 1 homogeneous p laplacian equations we consider a nonlinear elliptic equation driven by the p laplacian and with a reaction term which exhibits a p 1 homogeneous growth both near and near zero using critical point theory with truncation techniques the method of upper lower solutions and morse theory we show that the problem has five nontrivial smooth solutions four of which have constant sign two positive and two negative

eigenvalues of the p x laplacian steklov problem abstract consider steklov eigenvalue problem involving the p x laplacian on a bounded domain the open subset of r n with n 2 as follows p x u u p x 2 u in u p x 2 u u p x 2 u on where p x constant we prove that the existence of infinitely many eigenvalue sequences unlike the p laplacian case for a variable exponent p x constant there does not exist a principal eigenvalue and the set of all eigenvalues is not closed under some assumptions finally we present some sufficient conditions for the infimum of all eigenvalues is zero and positive respectively

existence of three nontrivial solutions for asymptotically p linear noncoercive p laplacian equations abstract we consider a nonlinear elliptic problem driven by the p laplacian where the right hand side nonlinearity exhibits a p linear behavior near infinity and the euler functional of the problem need not be coercive and in fact can be indefinite using a combination of minimax arguments with truncation techniques and morse theory we show that the problem has at least three nontrivial smooth solutions two of which have a constant sign our method of proof uses some results on critical groups and the spectrum of the p laplacian due to perera and dancer perera

numerical methods for computing nonlinear eigenpairs part i iso homogeneous cases with a rayleigh quotient formulation a local minimax method is developed to solve a class of iso homogeneous nonlinear eigenpair problems for multiple solutions in banach spaces following their instability order the algorithm is implemented to compute weighted eigenpairs of the p laplacian numerical eigenfunctions are illustrated by their graphics several interesting phenomena have been observed and are open for further investigation mathematical analysis related to convergence and instability order of computed eigenfunctions etc is also presented

multiplicity of nontrivial solutions for quasilinear elliptic equation two nontrivial solutions are obtained by the reduction method for the nonautonomous semilinear elliptic dirichlet boundary value problem some well known multiplicity results are generalized

resonance problems for the p laplacian systems abstract two existence theorems of the solutions are obtained for the p laplacian systems at resonance under a landesman lazer type condition by critical point theory

perturbation of the p laplacian by vanishing nonlinearities in one dimension abstract we treat the quasi linear spectral problem p u p 2 u u p 2 u h x u 0 x a u 0 u a 0 at resonance with vanishing nonlinearity h x u g u f x g u 0 as u and f l 0 a f 0 satisfying certain orthogonality related hypotheses 1 p 3 p 2 the parameter r takes an arbitrary resonant value we first establish the boundedness of the set of all weak solutions in the sobolev space w 0 1 p 0 a which then enables us to obtain an existence result by the leray schauder degree theory the boundedness is obtained from a very precise asymptotic estimate valid for large solutions to p which can be applied thanks to a sufficiently fast rate of decay g u 0 as u

perturbations of determinants of matrices abstract let a a be n n matrices and d k be an arbitrary orthonormal basis in c n it is proved that the determinants satisfy the inequality det a det a q k 1 n 1 1 2 a a d k a a d k where is the euclidean norm and q max k a a d k

multiple solutions for elliptic systems with nonlinearities of arbitrary growth abstract we prove the existence of infinitely many solutions for symmetric elliptic systems with nonlinearities of arbitrary growth moreover if the symmetry of the problem is broken by a small enough perturbation term we find at least three solutions the proofs utilise a variational setting given by de figueiredo and ruf in order to prove an existence s result and the algebraic approach based on the pohozaev s fibering method

on superquadratic elliptic systems in this article we study the existence of solutions for the elliptic system updelta u frac partial h partial v left u v x right text in upomega updelta v frac partial h partial v left u v x right text in upomega u 0 v 0 text on partial upomega where upomega is a bounded open subset of mathbb r n with smooth boundary partial upomega and the function h mathbb r 2 times overline upomega to mathbb r is of class c 1

resonance problems for the p laplacian with a nonlinear boundary condition abstract consider resonance problems at an arbitrary eigenvalue of the elliptic equations p u u p 2 u g 1 u h 1 x in u p 2 u u p 2 u g 2 u h 2 x on some existence theorems are obtained under the landesman lazer type conditions by using variational method

superlinear problems without ambrosetti and rabinowitz growth condition superlinear elliptic boundary value problems without ambrosetti and rabinowitz growth condition are considered existence of nontrivial solution result is established by combining some arguments used by struwe and tarantello and schechter and zou also by wang and wei firstly by using the mountain pass theorem due to ambrosetti and rabinowitz is constructed a solution for almost every parameter by varying the parameter then it is considered the continuation of the solutions

nonuniformly nonlinear elliptic equations of p biharmonic type this paper deals with the existence and multiplicity of weak solutions to nonlinear differential equations involving a general p biharmonic operator in particular p biharmonic operator under dirichlet boundary conditions or navier boundary conditions our method is mainly based on variational arguments

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

bounds for determinants of meet matrices associated with incidence functions abstract we consider meet matrices on meet semilattices as an abstract generalization of greatest common divisor gcd matrices some new bounds for the determinant of meet matrices and a formula for the inverse of meet matrices are given

variational elliptic problems which are nonquadratic at infinity p co if n 1 2 then it is well known that the weak solutions u e h 2 of p are the critical points of the c functional j u ulvul dx jcf x u dx where f x s jif x t dt in this variational setting the literature usually distinguishes between two situations for prob lem p the subquadratic situation where the potential f satisfies lim suplsl f x s s2 i c m and the superquadratic situation where f satisfies limlsl m f x s s2 00 the main goal of this paper is to present a unified approach to both situations by means of a condition of nonquadraticity at infinity on f more precisely our hypotheses are as follows

nontrivial solutions of superlinear p laplacian equations we consider p laplacian equations on a bounded domain where the nonlinearity is superlinear but dose not satisfy the usual ambrosetti rabinowitz condition near infinity or its dual version near zero nontrivial solutions are obtained by computing the critical groups and morse theory

resonance problems for the p laplacian abstract we consider resonance problems at an arbitrary eigenvalue of the p laplacian and prove the existence of weak solutions assuming a standard landesman lazer condition we use variational arguments to characterize certain eigenvalues and then to establish the solvability of the given boundary value problem

a class of adaptive control problems solved via stochastic control abstract following a set up investigated by rishel 7 we consider an adaptive control problem with unknown parameter x as a partially observed stochastic control problem exploiting the finite dimensionality of the estimator we transform it to a fully observed stochastic optimal control problem to which we then find optimal randomized feedback policies

doubly diagonally dominant matrices we consider the class of doubly diagonally dominant matrices a ajj e c la 1 l jjl ck i laiklck jlajkl i j and its subclasses we give necessary and sufficient conditions in terms of the directed graph for an irreducibly doubly diagonally dominant matrix to be a singular matrix or to be an h matrix as in the case of diagonal dominance we show that the schur complements of doubly diagonally dominant matrices inherit this property moreover we describe when a schur complement of a strictly doubly diagonally dominant matrix is strictly diagonally dominant 0 elsevier science inc 1997 1 preliminaries

existence of solutions for a class of noncooperative elliptic systems we establish the existence of a nontrivial solution for a class of noncooperative elliptic systems with nonlinearities of superlinear growth moreover if there is resonance we also find at least a solution all results are obtained by the minimax methods in critical point theory

an exact formula for a linear quadratic adaptive stochastic optimal control law for a linear quadratic adaptive stochastic optimal control problem a formula for the optimal control is obtained by applying variational methods to an equivalent problem obtained from the girsanov transformation the formula does depend on a quantity defined through a martingale representation of the conditional remaining cost

on a partially observable lqg problem for systems with markovian jumping parameters in this paper a variant of the lqg problem for systems with markovian jumping parameters is studied it is assumed that the system is partially observable in the sense that we do not have access to the jumping parameter via nonlinear filtering the problem is recast as one with complete information it is then shown that a certain averaged control is an optimal policy vis a vis a modified quadratic cost apart from its intrinsic theoretical interest this result has an important bearing on a number of different analyses related with the original quadratic cost

differential systems with strongly indefinite variational structure abstract the euler lagrange equations of lagrangians with a strongly indefinite quadratic part are studied by means of the direct min max method of benci and rabinowitz the simplest case is a system of two coupled semilinear poisson equations using an analytic framework of a suitable family of products of fractional sobolev spaces existence results are obtained for all nonlinearities with subcritical growth rates

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

the behavior of the best sobolev trace constant and extremals in thin domains in this paper we study the asymptotic behavior of the best sobolev trace constant and extremals for the immersion w 1 p o l q o in a bounded smooth domain when it is contracted in one direction we find that the limit problem when rescaled in a suitable way is a sobolev type immersion in weighted spaces over a projection of o w 1 p p o a l q p o b for the special case p q this problem leads to an eigenvalue problem with a nonlinear boundary condition we also study the convergence of the eigenvalues and eigenvectors in this case