existence of solutions for a p x kirchhoff type equation this paper is concerned with the existence and multiplicity of solutions to a class of p x kirchhoff type problem with dirichlet boundary data by means of a direct variational approach and the theory of the variable exponent sobolev spaces we establish conditions ensuring the existence and multiplicity of solutions for the problem

multiplicity of positive solutions for the kirchhoff type equations with critical exponent in rn abstract in this work we study the existence and multiplicity of solutions to the following kirchhoff type problem with critical nonlinearity in r n a b r n u p d x p u u p 1 f x u x r n u d 1 p r n where n 2 p a b 0 and the nonlinearity f x u satisfies certain subcritical growth conditions by using topological and variational methods infinitely many positive solutions are obtained

ground state of kirchhoff type fractional schrodinger equations with critical growth abstract in this paper we study the critical kirchhoff type fractional schrodinger equation 0 1 1 r 3 r 3 u x u y 2 x y 3 2 s d x d y s u u f u u 2 s 1 in r 3 where s 0 1 and 2 s 6 3 2 s we establish the pohoz ev type identity of 0 1 when s 3 4 1 under some conditions on and f u we obtain some results on the existence of ground state solutions when s 0 3 4 we also prove the non existence result in particular when 0 we obtain an existence result

existence of ground states for a kirchhoff type problem without 4 superlinear condition abstract we consider the existence of ground state solutions for the kirchhoff type problem a b r n u 2 d x u v x u u p 2 u x r n u h 1 r n where a b 0 n 1 2 3 and 2 p 2 here we are interested in the case that 2 p 4 since the existence of ground state for 4 p 2 is easily obtained by a standard variational argument our method is based on a poho z aev type identity

ground state sign changing solutions for fractional kirchhoff equations in bounded domains we study the existence of ground state sign changing solutions for the fractional kirchhoff problem under mild assumptions on the nonlinearity by using some new analytical skills and the non nehari manifold method we prove that the fractional kirchhoff problem possesses a ground state sign changing solution ub moreover we show that the energy of ub is strictly larger than twice that of the ground state solutions of nehari type finally we establish the convergence property of ub as the parameter b 0 our results generalize some results obtained by shuai j differ equations 259 1256 2015 and tang and cheng j differ equations 261 2384 2016

positive solutions for kirchhoff type equations with critical exponent in rn abstract in this paper we study the following nonlinear kirchhoff type equation 0 1 a b r n u 2 d x u u 2 2 u h x x r n u d 1 2 r n where a 0 b 0 n 3 2 2 n n 2 0 and h l 2 2 1 r n 0 is nonnegative by using the variational method we obtain the existence of positive solutions for eq 0 1 under some assumptions on a b

existence and multiplicity of solutions for a superlinear kirchhoff type equations with critical sobolev exponent in r n abstract in this article we consider a class of superlinear kirchhoff type equations with critical growth a b r n u 2 d x u u 2 1 k x u q 1 x r n u d 1 2 r n where 0 n 4 2 q 2 2 2 n n 2 a b 0 and a b 0 k satisfies some conditions by using the variational method the multiplicity of positive solutions and infinitely many pairs of solutions are obtained

stationary kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity this paper deals with the existence and the asymptotic behavior of non negative solutions for a class of stationary kirchhoff problems driven by a fractional integro differential operator lk and involving a critical nonlinearity in particular we consider the problem m u 2 lku f x u u 2s 2uin u 0in rn where rn is a bounded domain 2s is the critical exponent of the fractional sobolev space hs rn the function f is a subcritical term and is a positive parameter the main feature as well as the main difficulty of the analysis is the fact that the kirchhoff function m could be zero at zero that is the problem is degenerate the adopted techniques are variational and the main theorems extend in several directions previous results recently appeared in the literature

degree constrained 2 partitions of graphs a k1 k2 partition of a graph g is a vertex partition v1 v2 of g satisfying that g vi ki for i 1 2 we determine for all positive integers k1 k2 the complexity of deciding whether a given graph has a k1 k2 partition we also address the problem of finding a function g k1 k2 such that the k1 k2 partition problem is n p complete for the class of graphs of minimum degree less than g k1 k2 and polynomial for all graphs with minimum degree at least g k1 k2 we prove that g 1 k k for k 3 that g 2 2 3 and that g 2 3 if it exists has value 4 or 5

multiplicity results for the non homogeneous fractional p kirchhoff equations with concave convex nonlinearities in this paper we are interested in the multiplicity of solutions for a non homogeneous p kirchhoff type problem driven by a non local integro differential operator as a particular case we deal with the following elliptic problem of kirchhoff type with convex concave nonlinearities a b r 2 n u x u y p x y n s p d x d y 1 p s u 1 x u q 2 u 2 x u r 2 u h x in r n where p s is the fractional p laplace operator a b 0 with a b r 0 0 is a real parameter 0 s 1 p with sp r n under some suitable conditions we obtain the existence of two non trivial entire solutions by applying the mountain pass theorem and ekeland s variational principle a distinguished feature of this paper is that a may be zero which means that the above mentioned problem is degenerate to the best of our knowledge our results are new even in the laplacian case

on ground states for the kirchhoff type problem with a general critical nonlinearity abstract in this paper we consider the following kirchhoff type problem a b r 3 u 2 u f u in r 3 u h 1 r 3 u 0 in r 3 where a b 0 are constants and f has a critical growth the aim of this paper is to study the existence of ground state solutions for kirchhoff type equations with a general nonlinearity in the critical growth without the assumption of the monotonicity of the function t f t t 3 moreover we will show that the mountain pass value gives the least energy level and also obtain a mountain pass solution

existence and symmetry of solutions for critical fractional schrodinger equations with bounded potentials abstract this paper is concerned with the following fractional schrodinger equations involving critical exponents u v x u k x f u u 2 2 u in r n where is the fractional laplacian operator with 0 1 n 2 is a positive real parameter and 2 2 n n 2 is the critical sobolev exponent v x and k x are positive and bounded functions satisfying some extra hypotheses based on the principle of concentration compactness in the fractional sobolev space and the minimax arguments we obtain the existence of a nontrivial radially symmetric weak solution for the above mentioned equations without assuming the ambrosetti rabinowitz condition on the subcritical nonlinearity

the existence of least energy nodal solutions for some class of kirchhoff equations and choquard equations in r n abstract in this paper we study the existence of least energy nodal solutions for a class of kirchhoff type problems in r n since the kirchhoff equation is a nonlocal one the variational setting to look for sign changing solutions is different from the local cases by using constrained minimization on the sign changing nehari manifold we prove the kirchhoff problem has a least energy nodal solution with its energy exceeding twice the least energy as a co product of our approaches we obtain the existence of least energy sign changing solution for choquard equations in r n and show that the sign changing solution has an energy strictly larger than the least energy and less than twice the least energy

the brauer group of graded continuous trace c algebras let x be a locally compact hausdorff space the graded morita equivalence classes of separable z2 graded continuous trace c algebras which have spectrum x form a group gbr x the infinite dimensional graded brauer group of x techniques from algebraic topology are used to prove that gbr x is isomorphic via an isomorphism w to the direct sum t1 x z2 e 13 x z the group gbr x includes as a subgroup the ungraded continuous trace c algebras and the dixmier douady invariant of such an ungraded c algebra is its image in h3 x z under w introduction the study of graded c algebras has become particularly important since g g kasparov s development of kk theory for operator algebras 18 in this paper separable z2 graded continuous trace c algebras are classified the graded morita equivalence classes of such algebras whose spectra are all the same locally compact hausdorff space x form a group called the infinitedimensional graded brauer group of x and denoted by gbr x two invariants defined on gbr x provide useful insights into the structure of these c algebras and relate the results presented here to previous work the constructions of j dixmier and a douady 3 4 51 form an important framework for the graded classification let x be a locally compact hausdorff space with countable base dixmier and douady considered separable stable continuous trace c algebras with spectrum x there is a canonical way to associate such an algebra a with a fiber bundle a over x with fiber x x the compact operators on an infinite dimensional separable hilbert space let 7 w r be the projective unitary group of and let fi x g denote the tech cohomology of x with coefficients in the sheaf of germs of continuous functions from x to g for g a group then the isomorphism class of a is an element of fi x 37 x which can be shown to be isomorphic to f3i x z they defined the dixmier douady invariant 6 a e h13 x z of the algebra a and proved that the invariant defines a one to one correspondence between isomorphism classes of such algebras and the elements of h3 x z consider now the collection of graded separable continuous trace c algebras all with spectrum x we will define gbr x as the set of equivalence classes of all such c algebras under graded morita equivalence which is the graded version of strong morita equivalence defined by m rieffel 22 23 it is important to note received by the editors december 15 1986 1980 mathematics subject classification 1985 revision primary 46m20 46l80 46l35 secondary 55r10

nonlocal kirchhoff problems extinction and non extinction of solutions abstract in this paper we discuss the extinction and non extinction properties of solutions for the following fractional p kirchhoff problem u t m u s p p p s u u r 2 u u q 2 u x t 0 u 0 x t r n 0 u x 0 u 0 x x where u s p is the gagliardo seminorm of u r n is a bounded domain with lipschitz boundary p s is the fractional p laplacian with 0 s 1 p 2 m 0 0 is a continuous function 1 q 2 r 1 and 0 under suitable assumptions we obtain the extinction of solutions to get more precisely decay estimates of solutions we develop the gagliardo nirenberg inequality moreover the non extinction property of solutions is also investigated

on some critical problems for the fractional laplacian operator abstract we study the effect of lower order perturbations in the existence of positive solutions to the following critical elliptic problem involving the fractional laplacian 2 u u q u n n u 0 in u 0 on where r n is a smooth bounded domain n 1 0 0 q n n 0 min n 2 for suitable conditions on depending on q we prove in the case q 1 there exist at least two solutions for every 0 and some 0 at least one if no solution if for q 1 we show existence of at least one solution for 0 1 and nonexistence for 1 when q 1 the existence is shown for every 0 also we prove that the solutions are bounded and regular

positive solutions for a quasilinear elliptic equation of kirchhoff type this paper is concerned with the existence of positive solutions to the class of nonlocal boundary value problems of the type m w u 2dx du f x u email protected u 0 email protected email protected where w is a smooth bounded domain of n m is a positive function and f has subcritical growth

low regularity global solutions for nonlinear evolution equations of kirchhoff type we study the global solvability of the cauchy dirichlet problem for two second order in time nonlinear integro differential equations 1 the extensible beam plate equation utt 2u m u 2dx u 0 x t r 2 a special case of the kirchhoff equation utt a b u 2dx 2 u 0 x t r by exploiting the i method of j colliander m keel g staffilani h takaoka t tao we prove that both equations admit global in time infinite energy solutions in case 1 the energy is the mechanical energy in case 2 it is a second order invariant introduced by s i pokhozhaev for the extensible beam equation 1 we also consider the effect of linear dissipation on such low regularity solutions and we prove their exponential decay as t

infinitely many radial solutions for kirchhoff type problems in rn abstract in this paper we concern with a class of kirchhoff type problems in r n by using the fountain theorem we obtain three existence results of infinitely many radial solutions for the problem

the existence of a ground state solution for a class of kirchhoff type equations in ℝn in this paper we study the following kirchhoff type equation where a b are positive constants and n 1 2 3 under appropriate assumptions on v k and g we obtain a ground state solution by using the approach developed by szulkin and weth in 2010

existence and multiplicity of nontrivial solutions for a class of schrodinger kirchhoff type equations abstract in the present paper the following schrodinger kirchhoff type problem p a b r n u 2 d x u v x u f x u in r n u x 0 as x is studied when n 2 3 4 and v x 1 the existence theorem of nontrivial weak solutions for problem p is obtained when n 1 2 3 and v x is more general two existence theorems of nontrivial weak solutions and a sequence of high energy weak solutions for problem p are obtained

existence of solutions for a bi nonlocal fractional p kirchhoff type problem in this paper we are concerned with the existence of nonnegative solutions for a p kirchhoff type problem driven by a non local integro differential operator with homogeneous dirichlet boundary data as a particular case we study the following problem m x u s p p p s u f x u u s p p in u 0 in r n u s p p r 2 n u x u y p x y n p s d x d y where p s is a fractional p laplace operator is an open bounded subset of r n with lipschitz boundary m r 0 r is a continuous function and f r r 0 r is a continuous function satisfying the ambrosetti rabinowitz type condition the existence of nonnegative solutions is obtained by using the mountain pass theorem and an iterative scheme the main feature of this paper lies in the fact that the kirchhoff function m depends on x and the nonlinearity f depends on the energy of solutions

on critical point theory for indefinite functionals in the presence of symmetries we consider functionals which are not bounded from above or from below even modulo compact perturbations and which exhibit certain symmetries with respect to the action of a compact lie group we develop a method which permits us to prove the existence of multiple critical points for such functionals the proofs are carried out directly in an infinite dimensional hilbert space and they are based on minimax arguments the applications given here are to hamiltonian systems of ordinary differential equations where the existence of multiple time periodic solutions is established for several classes of hamiltonians symmetry properties of these hamiltonians such as time translation invariancy or evenness are exploited introduction many variational problems which arise from physics or mathematical physics are indefinite in the sense that the functionals involved are not bounded from below or from above however some of these functionals defined in an appropriate function space are semidefinite in the sense that they are bounded from below or from above if perturbed with a weakly continuous functional this paper deals with functionals which are not semidefinite usually problems involving indefinite functionals are more difficult to handle and only very recently a method has been developed which permits us to treat such problems directly in an infinite dimensional function space 7 in 7 some theorems have been proved which establish the existence of at least one nontrivial critical point for such functionals in many physical situations there are problems which have symmetries with respect to the action of some lie group in this case we expect the existence of many critical points this has been established for semidefinite functionals cf e g 8 and 3 for even functionals 9 for a zp action with p prime numbers 15 16 10 and 6 for an s action in this paper we develop a method which allows us to estimate the number of critical points of indefinite functionals in the presence of symmetries more precisely received by the editors july 16 1980 and in revised form june 26 1981 1980 mathematics subject classification primary 58e05 70h05 secondary 34c25

the brezis nirenberg result for the fractional laplacian the aim of this paper is to deal with the non local fractional counterpart of the laplace equation involving critical non linearities studied in the famous paper of brezis and nirenberg 1983 namely our model is the equation su u u 2 2u in u 0 in rn where s is the fractional laplace operator s 0 1 is an open bounded set of rn n 2s with lipschitz boundary 0 is a real parameter and 2 2n n 2s is a fractional critical sobolev exponent in this paper we first study the problem in a general framework indeed we consider the equation lku u u 2 2u f x u 0 in u 0 in rn where lk is a general non local integrodifferential operator of order s and f is a lower order perturbation of the critical power u 2 2u in this setting we prove an existence result through variational techniques then as a concrete example we derive a brezis nirenberg type result for our model equation that is we show that if 1 s is the first eigenvalue of the non local operator s with homogeneous dirichlet boundary datum then for any 0 1 s there exists a non trivial solution of the above model equation provided n 4s in this sense the present work may be seen as the extension of the classical brezis nirenberg result to the case of non local fractional operators

infinitely many positive solutions for kirchhoff type problems abstract this paper is concerned with the existence of infinitely many positive solutions to a class of kirchhoff type problem a b u 2 d x u f x u in and u 0 on where is a smooth bounded domain of r n a b 0 0 and f r r is a caratheodory function satisfying some further conditions we obtain a sequence of a e positive weak solutions to the above problem tending to zero in l with f being more general than that of k perera z zhang nontrivial solutions of kirchhoff type problems via the yang index j differential equations 221 2006 246 255 z zhang k perera sign changing solutions of kirchhoff type problems via invariant sets of descent flow j math anal appl 317 2006 456 463

sign changing solutions for a fractional kirchhoff equation abstract using a minimization argument and a quantitative deformation lemma we establish the existence of least energy sign changing solutions for the following nonlinear kirchhoff problem a b u 2 s u v x u k x f u in r 3 where a b 0 are constants s 0 1 s is the fractional laplacian v k are continuous positive functions allowed for vanishing behavior at infinity and f is a continuous function satisfying suitable growth assumptions moreover when the nonlinearity f is odd we obtain the existence of infinitely many nontrivial weak solutions not necessarily nodals

on the fractional schrodinger kirchhoff equations with electromagnetic fields and critical nonlinearity abstract in this paper we consider the fractional schrodinger kirchhoff equations with electromagnetic fields and critical nonlinearity e 2 s m u s a e 2 a e s u v x u u 2 s 2 u h x u 2 u x r n u x 0 as x where a e s is the fractional magnetic operator with 0 s 1 2 s 2 n n 2 s m r 0 r is a continuous nondecreasing function v r n r 0 and a r n r n are the electric and magnetic potentials respectively by using the fractional version of the concentration compactness principle and variational methods we show that the above problem i has at least one solution provided that e e and ii for any m n has m pairs of solutions if e e m where e and e m are sufficiently small positive numbers moreover these solutions u e 0 as e 0

on a model connected with the kirchhoff equation abstract we prove a global existence result in suitable sobolev spaces for the solution of the nonlinear cauchy problem tt u t x 1 r n dy k x y y u t y 2 x u t x 0 u 0 x u 0 x t u 0 x u 1 x 0 where n 3 is the space dimension the initial data are c with compact support in rn and k z is a positive smooth function rapidly decreasing as z our approach is based on the energy estimates combined with the classical von wahl inequalities

fractional kirchhoff type equation with hardy littlewood sobolev critical exponent abstract in this paper we consider the following fractional kirchhoff type equation a b r n r n u x u y 2 x y n 2 s d x d y s u i u 2 h u 2 h 2 u in r n where n 3 s 0 1 a 0 b 0 0 n and 2 h n n 2 s is the hardy littlewood sobolev critical exponent and i is the riesz potential we establish the existence nonexistence and multiplicity of nontrivial solutions to above equation

existence and asymptotic behavior of nodal solutions for the kirchhoff type problems in r3 abstract in this paper we study the existence and asymptotic behavior of nodal solutions to the following kirchhoff problem a b r 3 u 2 d x u v x u f x u in r 3 u h 1 r 3 where v x is a smooth function a b are positive constants because the so called nonlocal term r 3 u 2 d x u is involved in the equation the variational functional of the equation has totally different properties from the case of b 0 under suitable construction conditions we prove that for any positive integer k the problem has a sign changing solution u k b which changes signs exactly k times moreover the energy of u k b is strictly increasing in k and for any sequence b n 0 n there is a subsequence b n s such that u k b n s converges in h 1 r 3 to w k as s where w k also changes signs exactly k times and solves the following equation a u v x u f x u in r 3 u h 1 r 3

approximate solution of the backward problem for kirchhoff s model of parabolic type with discrete random noise abstract in this paper we consider the final boundary value problem for non local kirchhoff s model of parabolic type with discrete random noise we first discuss the instability of solutions then we present the regularized solution by the trigonometric method in non parametric regression associated with the truncated expansion method in addition under a prior assumption on the exact solution the convergence rate is obtained in the last section the effectiveness and suitability of our results are justified via numerical experiments

complexity and kernels for bipartition into degree bounded induced graphs in this paper we study the parameterized complexity of the problems of partitioning the vertex set of a graph into two parts v a and v b such that v a induces a graph with degree at most a resp an a regular graph and v b induces a graph with degree at most b resp a b regular graph these two problems are called upper degree bounded bipartition and regular bipartition respectively when a b 0 the two problems become the polynomially solvable problem of checking the bipartition of a graph when a 0 and b 1 regular bipartition becomes a well known np hard problem called dominating induced matching in this paper firstly we prove that the two problems are np complete with any nonnegative integers a and b except a b 0 secondly we show the fixed parameter tractability of these two problems with parameter k v a being the size of one part of the bipartition by deriving several problem kernels for them and constrained versions of them

graded brauer groups of a groupoid with involution abstract we define a group brr g containing in a sense the graded complex and orthogonal brauer groups of a locally compact groupoid g equipped with an involution when the involution is trivial we show that the new group naturally provides a generalisation of donovan karoubi s graded orthogonal brauer group gbro more generally it is shown to be a direct summand of the well known graded complex brauer group in addition we prove that brr g identifies with a direct sum of a real cohomology group and the abelian group extr g s 1 of real graded s 1 central extensions a cohomological picture is then given

existence and concentration result for the kirchhoff type equations with general nonlinearities in this paper we study the existence and concentration behaviors of positive solutions to the kirchhoff type equations varepsilon 2 m left varepsilon 2 n int mathbf r n nabla u 2 mathrm d x right delta u v x u f u quad rm in mathbf r n quad u in h 1 mathbf r n n geqq 1 where m and v are continuous functions under suitable conditions on m and general conditions on f we construct a family of positive solutions u varepsilon varepsilon in 0 tilde varepsilon which concentrates at a local minimum of v after extracting a subsequence ek

an approximate algorithm for a kirchhoff wave equation the initial boundary value problem for a hyperbolic type quasilinear equation w tt varphi int 0 pi w x 2dx w xx is considered its solution is derived by means of a numerical algorithm consisting of the projection method and the difference method for approximation with respect to spatial and time variables while the resulting discrete system is solved by an iteration process the accuracy of the proposed algorithm is estimated

deformation quantization for heisenberg supergroup we construct a non formal deformation machinery for the actions of the heisenberg supergroup analogue to the one developed by m rieffel for the actions of r d however the method used here differs from rieffel s one we obtain a universal deformation formula for the actions of r m n as a byproduct of weyl ordered kirillov s orbit method adapted to the graded setting to do so we have to introduce the notion of c superalgebra which is compatible with the deformation and which can be seen as corresponding to noncommutative superspaces we also use this construction to interpret the renormalizability of a noncom mutative quantum field theory

eigenvalue problems with unbalanced growth nonlinear patterns and standing wave solutions abstract we consider two classes of nonlinear eigenvalue problems with double phase energy and lack of compactness we establish existence and non existence results and related properties of solutions our analysis combines variational methods with the generalized pucci serrin maximum principle

existence and multiplicity of solutions for critical kirchhoff type p laplacian problems abstract in this paper we deal with the existence multiplicity and asymptotic behavior of solutions for the following stationary kirchhoff problems involving the p laplacian m u p d x p u f x u u p 2 u in u 0 on where is a bounded smooth domain of r n m r 0 r 0 is a continuous function satisfying some extra assumptions by using the mountain pass theorem the existence of solutions is obtained by applying an abstract critical point theorem based on the cohomological index we prove the existence of multiple solutions for the above problem when parameter belongs to some left neighborhood of the eigenvalue of the nonlinear operator u 1 p p where p n p p we also investigate the asymptotic behavior of solutions when converges to infinity the main feature as well as the main difficulty of our problem is the fact that the kirchhoff term m could be zero at zero i e the problem is degenerate

positive solutions for a critical p laplacian problem with a kirchhoff term abstract in this article the following p laplacian problem with kirchhoff term and critical exponent is studied a b r n u p d x p u u p 1 h x x r n u d 1 p r n where a 0 b 0 1 p n p n p n p p u d i v u p 2 u and h l p p 1 r n is nonzero and nonnegative when 0 with the help of the recent result of b sciunzi adv math 291 2016 12 23 infinitely many positive solutions are obtained by some analysis techniques while 0 by the variational method some results about positive solutions are obtained

existence of a positive solution for a kirchhoff problem type with critical growth via truncation argument abstract in this paper we shall show an existence result of a positive solution for a kirchhoff problem type in a bounded domain of r n that is for the problem m u 2 d x u f x u u 2 2 u in u 0 on we shall study the asymptotic behavior of this solution when converges to infinity our approach is based on the variational method an appropriated truncated argument and a priori estimates to obtain the solution

critical fractional p kirchhoff type problem with a generalized choquard nonlinearity in this article based on the concentration compactness principle and the mountain pass theorem we establish the existence of positive solutions to the fractional p kirchhoff type problem with a generalized choquard nonlinearity and a critical hardy sobolev term

multiple nontrivial solutions to a pp kirchhoff equation abstract in this paper morse theory is used to show the existence of nontrivial solutions to the nonlocal p kirchhoff type quasilinear elliptic equation m u p d x p 1 p u f x u in a smooth bounded domain r n with dirichlet boundary condition

degree constrained decompositions of graphs bounded treewidth and planarity we study the problem of decomposing the vertex set v of a graph into two nonempty parts v1 v2 which induce subgraphs where each vertex v v1 has degree at least a v inside v1 and each v v2 has degree at least b v inside v2 we give a polynomial time algorithm for graphs with bounded treewidth which decides if a graph admits a decomposition and gives such a decomposition if it exists this result and its variants are then applied to designing polynomial time approximation schemes for planar graphs where a decomposition does not necessarily exist but the local degree conditions should be met for as many vertices as possible

global nonexistence for abstract evolution equations with positive initial energy abstract in this paper we consider the problem of noncontinuation of solutions of the initial value problem for abstract evolution equations of the form pu tt q t u t a t u f t u t j 0 wherepandqare linear self adjoint operators anda t u andf t u are respectively a linear operator inu typically of differential type and a nonlinear driving force our principal concern is the noncontinuation or blow up of solutions when the initial energy is positive but appropriately bounded

global nonexistence for nonlinear kirchhoff systems in this paper we consider the problem of non continuation of solutions of dissipative nonlinear kirchhoff systems involving the p x laplacian operator and governed by nonlinear driving forces f f t x u as well as nonlinear external damping terms q q t x u u t both of which could significantly dependent on the time t the theorems are obtained through the study of the natural energy eu associated to the solutions u of the systems thanks to a new approach of the classical potential well and concavity methods we show the nonexistence of global solutions when the initial energy is controlled above by a critical value that is when the initial data belong to a specific region in the phase plane several consequences interesting in applications are given in particular subcases the results are original also for the scalar standard wave equation when p 2 and even for problems linearly damped

on a critical kirchhoff type problem abstract in the present paper we study a kirchhoff type problem involving the critical sobolev exponent we give sufficient conditions for the sequentially weakly lower semicontinuity and the palais smale property of the energy functional associated to the problem

multiplicity results for a p laplacian discrete problems of kirchhoff type this paper is concerned with the existence and multiplicity solutions for a discrete boundary value problem of p kirchhoff type our technical approach is based on variational methods

twisted k theory and poincare duality using methods of kk theory we generalize poincare k duality to the framework of twisted k theory

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

a critical kirchhoff type problem involving a nonlocal operator abstract in this paper we show the existence of non negative solutions for a kirchhoff type problem driven by a nonlocal integrodifferential operator that is m u z 2 l k u f x u u 2 2 u in u 0 in r n where l k is an integrodifferential operator with kernel k is a bounded subset of r n m and f are continuous functions z is a functional norm and 2 is a fractional sobolev exponent

sign changing solutions for a class of kirchhoff type problem in bounded domains abstract we are interested in the existence of least energy sign changing solutions for a class of kirchhoff type problem in bounded domains because the so called nonlocal term b u 2 d x u is involving in the equation the variational functional of the equation has totally different properties from the case of b 0 combining constraint variational method and quantitative deformation lemma we prove that the problem possesses one least energy sign changing solution u b moreover we show that the energy of u b is strictly larger than the ground state energy finally we regard b as a parameter and give a convergence property of u b as b 0

higher nonlocal problems with bounded potential abstract the aim of this paper is to study a class of nonlocal fractional laplacian equations depending on two real parameters more precisely by using an appropriate analytical context on fractional sobolev spaces due to servadei and valdinoci we establish the existence of three weak solutions for nonlocal fractional problems exploiting an abstract critical point result for smooth functionals we emphasize that the dependence of the underlying equation from one of the real parameters is not necessarily of affine type

multiple solutions for a fractional p kirchhoff problem with hardy nonlinearity abstract this paper is devoted to study the existence of multiple solutions for the following fractional p kirchhoff problem 0 1 m r 2 n u x u y p x y n p s d x d y p s u u q 2 u u r 2 u x in u 0 in r n where p s denotes the fractional p laplace operator is a smooth bounded set in r n containing 0 with lipschitz boundary m t a b t 1 with a 0 b 0 1 0 1 q p p r p p n p n p s is the fractional critical hardy sobolev exponent for 0 p s n by using fibering maps and nehari manifold we obtain that the existence of multiple solutions to problem 0 1 for both hardy sobolev subcritical and critical cases in particular the concentration compactness principle will be used to overcome the lack of compactness for the critical case

sign changing solutions of a class of nonlocal quasilinear elliptic boundary value problems abstract this paper is concerned with nonlocal kirchhoff s equation a b u 2 u f x u in u 0 on via variational methods and invariant sets of descent flow we obtain existence of signed and sign changing solutions with asymptotically 3 linear bounded nonlinearity

an equivariant brauer group and actions of groups onc algebras suppose that g t is a second countable locally compact trans formation group given by a homomorphism l g homeo t and that a is a separable continuous trace c algebra with spectrum t an action g aut a is said to cover l if the induced action of g on t coincides with the original one we prove that the set brg t of morita equivalence classes of such systems forms a group with multiplication given by the balanced tensor product a b a c0 t b and we refer to brg t as the equivariant brauer group we give a detailed analysis of the structure of brg t in terms of the moore cohomology of the group g and the integral cohomology of the space t using this we can characterize the stable continuous trace c algebras with spectrum t which admit actions covering l in particular we prove that if g r then every stable continuous trace c algebra admits an essentially unique action covering l thereby substantially improving results of raeburn and rosenberg

multiplicity results for stationary kirchhoff problems involving fractional elliptic operator and critical nonlinearity in rn in this paper we study a class of stationary kirchhoff problems involving a fractional elliptic operator and critical nonlinearity in rn g u s2 su k x u q 2u u 2s 2u by using a fractional version of lions second concentration compactness principle and concentration compactness principle at infinity to prove that the ps c condition holds locally and by minimax methods and krasnoselskii genus theory we establish the multiplicity of solutions for suitable positive parameters

on the kernelization of split graph problems abstract a split graph is a graph whose vertices can be partitioned into a clique and an independent set we study numerous problems on split graphs namely the k vertex disjoint paths k cycle k path and k l stable set problems in the k vertex disjoint paths problem we are given a graph and k terminal pairs of vertices and are asked whether there is a set of k vertex disjoint paths linking these terminal pairs respectively in the k cycle k path problem we are given a graph and are asked whether there is a path cycle of length k the k l stable set problem takes a graph and an integer k as input and asks whether the graph has a subset of k vertices such that the distance between every two vertices in the subset is at least l 1 it is known that all the above problems are np complete on split graphs we derive a 4 k vertex kernel for the k vertex disjoint paths problem and an o k 2 vertex kernel for both the k path problem and the k cycle problem concerning the k l stable set problem for l 1 or l 3 the problem is polynomial time solvable on split graphs for l 2 we prove that the k l stable set problem is w 1 complete on split graphs with respect to k however if the given split graph contains no k 1 r as an induced subgraph and every vertex in the independent set of the split graph has degree at most d we derive a linear vertex kernel for the k 2 stable set problem where both r and d are constants

existence of positive ground state solutions for kirchhoff type problems abstract in this paper we study the existence of positive ground state solutions for the nonlinear kirchhoff type problem a b r 3 u 2 u v x u f u in r 3 u h 1 r 3 u 0 in r 3 where a b 0 are constants f c r r is subcritical near infinity and superlinear near zero and satisfies the berestycki lions condition by using an abstract critical point theorem established by jeanjean and a new global compactness lemma we show that the above problem has at least a positive ground state solution our result generalizes the results of li and ye 2014 concerning the nonlinearity f u u p 1 u with p 2 5

on nonlocal p x laplacian dirichlet problems abstract this paper deals with the nonlocal p x laplacian dirichlet problems with non variational form a u p x u x b u f x u x in u 0 and with variational form a u p x p x d x p x u x b f x u d x f x u x in u 0 where f x t 0 t f x s d s and a is allowed to be singular at zero using s mapping theory and the variational method some results on existence and multiplicity for the problems are obtained under weaker hypotheses our results are also new even for the case when p x p is a constant

sign changing solutions of kirchhoff type problems via invariant sets of descent flow we obtain sign changing solutions of a class of nonlocal quasilinear elliptic boundary value problems using variational methods and invariant sets of descent flow

sign changing and multiple solutions of kirchhoff type problems without the p s condition abstract the existence of nontrivial solutions of kirchhoff type equations is an important nonlocal quasilinear problem in this paper we use minimax methods and invariant sets of descent flow to prove two interesting existence theorems for the following 4 superlinear kirchhoff type problems without the p s condition one concerning the existence of a nontrivial solution and the other one concerning the existence of sign changing solutions and multiple solutions a b u 2 u f x u in u 0 on

multiplicity of solutions for p x polyharmonic elliptic kirchhoff equations abstract in this paper we establish the existence of an unbounded sequence of solutions for a class of quasilinear elliptic p x polyharmonic kirchhoff equations including the new delicate degenerate case not yet covered in the literature the main tool is the symmetric mountain pass theorem of ambrosetti and rabinowitz

multiple and nodal solutions for parametric neumann problems with nonhomogeneous differential operator and critical growth abstract we consider a parametric neumann problem with nonhomogeneous differential operator and critical growth combining variational methods based on critical point theory with suitable truncation techniques and flow invariance arguments we show that for all large the problem has at least three nontrivial smooth solutions two of constant sign one positive the other negative and the third nodal we also study the asymptotic behavior of all solutions obtained when converges to infinity the interesting point is that we do not impose any restrictions to the behavior of the nonlinear term f at infinity our work unifies and sharply improves several recent papers

fixed parameter tractability and completeness ii on completeness for w 1 abstract for many fixed parameter problems that are trivially solvable in polynomial time such as k dominating set essentially no better algorithm is presently known than the one which tries all possible solutions other problems such as feedback vertex set exhibit fixed parameter tractability for each fixed k the problem is solvable in time bounded by a polynomial of degree c where c is a constant independent of k in a previous paper the w hierarchy of parameterized problems was defined and complete problems were identified for the classes w t for t 2 our main result shows that independent set is complete for w 1

existence of solutions for kirchhoff type problem involving the non local fractional p laplacian abstract the purpose of this paper is to investigate the existence of weak solutions for a kirchhoff type problem driven by a non local integro differential operator of elliptic type with homogeneous dirichlet boundary conditions as follows m r 2 n u x u y p k x y d x d y l k p u f x u in u 0 in r n where l k p is a non local operator with singular kernel k is an open bounded subset of r n with lipshcitz boundary m is a continuous function and f is a caratheodory function satisfying the ambrosetti rabinowitz type condition we discuss the above mentioned problem in two cases when f satisfies sublinear growth condition the existence of nontrivial weak solutions is obtained by applying the direct method in variational methods when f satisfies suplinear growth condition the existence of two nontrivial weak solutions is obtained by using the mountain pass theorem

existence and multiplicity of solutions for fractional schrodinger kirchhoff equations with trudinger moser nonlinearity abstract we study the existence and multiplicity of solutions for a class of fractional schrodinger kirchhoff type equations with the trudinger moser nonlinearity more precisely we consider m u n s n s s u v x u n s 1 u f x u h x u p 2 u in r n u r 2 n u x u y n s x y 2 n d x d y r n v x u n s d x s n where m 0 0 is a continuous function s 0 1 n 2 0 is a parameter 1 p n s s is the fractional n s laplacian v r n 0 is a continuous function f r n r r is a continuous function and h r n 0 is a measurable function first using the mountain pass theorem a nonnegative solution is obtained when f satisfies exponential growth conditions and is large enough and we prove that the solution converges to zero in w v s n s r n as then using the ekeland variational principle a nonnegative nontrivial solution is obtained when is small enough and we show that the solution converges to zero in w v s n s r n as 0 furthermore using the genus theory infinitely many solutions are obtained when m is a special function and is small enough we note that our paper covers a novel feature of kirchhoff problems that is the kirchhoff function m 0 0