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

nontrivial solutions of kirchhoff type problems via the yang index abstract we obtain nontrivial solutions of a class of nonlocal quasilinear elliptic boundary value problems using the yang index and critical groups

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

theta caracteristiques des courbes tracees sur une surface lisse nous generalisons la notion de theta caracterisque sur une courbe lisse a une courbe de gorenstein puis construisons un espace de modules t au sens grossier pour les theta caracteristiques des courbes de degre d sur une surface algebrique projective lisse nous montrons que le schema t est projectif et determinons l ensemble de ses points fermes ensuite nous generalisons le theoreme d atiyah et mumford sur l invariance sous deformation de la dimension modulo 2 de l espace des sections globales d une theta caracterique a notre situation ce qui donne en corollaire que t est reunion de deux fermes disjoints puis nous donnons un critere de lissite et calculons l espace tangent en un point stable a t enfin nous considerons le cas ou la surface est le plan projectif nous montrons que t n a que des singularites rationnelles et calculons sa dimension ensuite nous decomposons t en comosantes irreductibles en utilisant la decomposition sur l ouvert des courbes lisses de beauville et catanese puis donnons une description generique de ces composantes irreductibles

existence of solution for a class of nonvariational kirchhoff type problem via dynamical methods abstract in this paper we use dynamical methods to establish the existence of nontrivial solution for the following class of kirchhoff type problem p 1 b u 2 d x u f u h u u x u 0 x where r n n 2 3 is a smooth bounded domain b 0 f r r and h r r n 0 are c 1 functions that satisfy some technical conditions

on ground states of superlinear p laplacian equations in rn abstract we consider superlinear p laplacian equations in r n with a potential which is periodic or has a bounded potential well without assuming the ambrosetti rabinowitz type condition and the monotonicity of the function t f x t t p 1 we prove the existence of ground states even for the case p 2 our results extend the recent results of li wang and zeng ann inst h poincare anal non lineaire 23 2006 829 837

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

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

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

cohomologie des varietes de modules de hauteur nulle soient r c1 c2 des entiers avec r 1 m r c 1 c 2 la variete de modules des faisceaux algebriques semi stables sur 2 de rang r et de classes de chern c 1 c 2 et m s r c 1 c 2 l ouvert de m r c 1 c 2 correspondant aux faisceaux stables on note la pente et le discriminant des faisceaux de rang r et de classes de chern c 1 c 2 il existe une unique fonction telle que pour tout choix de r c 1 c 2 m r c 1 c 2 est de dimension positive si et seulement si si on dit que la variete de modules m r c 1 c 2 est de hauteur nulle soit la caracteristique d euler poincare des faisceaux coherents de rang r et de classes de chern c 1 c 2 on suppose que les entiers r c 1 et sont premiers entre eux cette condition signifie que m r c 1 c 2 m s r c 1 c 2 c est une variete projective lisse irreductible on etudie dans cet article la cohomologie entiere de m r c 1 c 2 pour cela on utilise une description particuliere de m r c 1 c 2 soient q m n des entiers avec q 2 m 0 n 0 les applications lineaires m q n sont appelees q modules de kronecker l espace projectif des droites de w l m q n est muni d une action evidente du groupe reductif g sl m sl n qui se prolonge a o 1 on note ss resp s l ouvert de des points semi stables resp stables pour cette action soit n q m n resp ns q m n le quotient algebrique de ss resp s par g il existe des entiers q m n tels que m r c 1 c 2 soit isomorphe a n q m n la condition r c 1 et sont premiers entre eux est equivalente a m et n sont premiers entre eux pour etudier la cohomologie de n q m n on utilise deux methodes la seconde uniquement dans un cas particulier la premiere methode est celle d atiyah bott qui a servi a etudier la cohomologie entiere des varietes de modules de fibres stables sur une courbe projective lisse le principal resultat obtenu est que la cohomologie entiere de m r c 1 c 2 est nulle en degre impair et sans torsion on trouve aussi des generateurs naturels de cette cohomologie la methode permet en outre de calculer les nombres de betti de n q m n le seconde methode utilise des resultats de bialynicki birula on ne traite que le cas de n 3 2 3 elle revient a etudier une action naturelle d un tore sur n 3 2 3 et a determiner ses points fixes on en deduit que n 3 2 3 possede une decomposition cellulaire

ranking primitive recursions the low grzegorczyk classes revisited traditional results in subrecursion theory are integrated with the recent work in predicative recursion by defining a simple ranking rho of all primitive recursive functions the hierarchy defined by this ranking coincides with the grzegorczyk hierarchy at and above the linear space level thus the result is like an extension of the schwichtenberg muller theorems when primitive recursion is replaced by recursion on notation the same series of classes is obtained except with the polynomial time computable functions at the first level

existence of solutions for perturbed fractional p laplacian equations abstract the purpose of this paper is to investigate the existence of weak solutions for a perturbed nonlinear elliptic equation driven by the fractional p laplacian operator as follows p s u v x u p 2 u a x u r 2 u b x u q 2 u in r n where is a real parameter p s is the fractional p laplacian operator with 0 s 1 p p r min q p s and v a b r n 0 are three positive weights using variational methods we obtain nonexistence and multiplicity results for the above mentioned equations depending on and according to the integrability properties of the ratio a q p b r p our results extend the previous work of autuori and pucci 2013 5 to the fractional p laplacian setting furthermore we weaken one of the conditions used in their paper hence the results of this paper are new even in the fractional laplacian case

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

poisson structures on moduli spaces of sheaves over poisson surfaces we introduce and study the notion of poisson surface we prove that the choice of a poisson structure on a surfaces canonically determines a poisson structure on the moduli space of stable sheaves ons this result generalizes previous results obtained by mukai 14 for abelian ork3 surfaces and by tyurin 16

viscosity solutions of fully nonlinear parabolic systems abstract in this paper we discuss the viscosity solutions of the weakly coupled systems of fully nonlinear second order degenerate parabolic equations and their cauchy dirichlet problem we prove the existence uniqueness and continuity of viscosity solution by combining perron s method with the technique of coupled solutions the results here generalize those in proc london math soc 63 1991 212 240 and comm partial differential equations 16 1991 1095 1128

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

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

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

mountain pass solutions for non local elliptic operators abstract the purpose of this paper is to study the existence of solutions for equations driven by a non local integrodifferential operator with homogeneous dirichlet boundary conditions these equations have a variational structure and we find a non trivial solution for them using the mountain pass theorem to make the nonlinear methods work some careful analysis of the fractional spaces involved is necessary we prove this result for a general integrodifferential operator of fractional type and as a particular case we derive an existence theorem for the fractional laplacian finding non trivial solutions of the equation s u f x u in u 0 in r n as far as we know all these results are new

groupe de picard des varietes de modules de fibres semi stables sur les courbes algebriques soient x une courbe algebrique projective lisse de genre g 2 sur r d des entiers avec r 2 on note u r d la variete de modules des fibres algebriques semi stables sur x de rang r et de degre d us r d l ouvert de u r d correspondant aux fibres stables on sait que u r d est une variete algebrique projective irreductible et normale si r et d ne sont pas premiers entre eux u r d n est pas lisse sauf dans une exception le cas ou g 2 r 2 et d est pair on supposera dans cet article qu on n est pas dans ce cas on a alors codimu r d u r d us r d 2 et u r d us r d est le lieu des points singuliers de u r d si l est un fibre en droites de degre d sur x on note u r l la sous variete fermee de u r d correspondant aux fibres vectoriels de determinant isomorphe a l on etudie dans cet article les groupes de picard pic u r d et pic u r l on montre d abord que les varietes u r d et u r l sont localement factorielles on calcule ensuite pic u r d et pic u r l qui sont decrits au moyen des diviseurs thetas generalises

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

separating nc along the δ axis the axis consists of certain classes nk such that uniform nc k nk a diagonalization is used to prove that nk nk 1 for all k 0 this separation of nc along the axis does not imply nck nck 1 because the relationship between nck and nk is not known however it is argued that the axis provides a natural subdivision of nc to support this a careful analogy with the classes dtime nk is made illuminating a surprisingly close relationship between ptime and nc the definition of nk uses ramified constructs related to a combination of parallel time and space

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

remarks on multiple nontrivial solutions for quasi linear resonant problems in this paper morse theory and local linking are used to study the existence of multiple nontrivial solutions for a class of dirichlet boundary value problems with double resonance at infinity and at 0

varietes de modules extremales de faisceaux semi stables sur ℙ 2 ℂ soient r c1 c2 des entiers avec r 1 et m r c1 c2 la variete de modules des faisceaux algebriques semi stables sur 2 de rang r et de classes de chern c1 c2 on dit que m r c1 c2 est extremale si dim m r c1 c2 0 et dim m r c1 c2 1 0 on se donne ici une description de certaines de ces varietes dites de faible hauteur si m r c1 c2 est de faible hauteur il existe des fibres exceptionnels e g f et des entiers q h n tels que tout faisceau semi stable e de rang r et de classes de chern c1 c2 sur 2 soit isomorphe au conoyau d un morphisme injectif de faisceaux f g q f h e n on considere l espace vectoriel w hom g q f h e n et l action evidente du groupe algebrique g aut g q f h aut e n sur w on definit une notion de semi stabilite pour l action de ce groupe en general non reductif et on montre qu en associant a un morphisme semi stable f le faisceau coker f on definit un morphisme w ss m r c1 c2 ou w ss est l ouvert g invariant des points semi stables qui est un bon quotient de m r c 1 c2 par g

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

twisted cubics on cubic fourfolds we construct a new twenty dimensional family of projective eight dimen sional irreducible holomorphic symplectic manifolds the compactified moduli space m3 y of twisted cubics on a smooth cubic fourfold y that does not contain a plane is shown to be smooth and to admit a contractionm3 y z y to a projective eight dimensional symplec tic manifoldz y the construction is based on results on linear determinantal representations of singular cubic surfaces

homoclinic solutions for fractional discrete laplacian equations abstract in this paper we introduce a variational framework to a fractional difference equation on z driven by the fractional discrete laplacian and involving a coercive weight function and a positive parameter by means of the critical point theory we prove the existence of at least two nontrivial and nonnegative homoclinic solutions for big enough

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

positive solutions for a system of second order quasilinear boundary value problems abstract in this paper we study the existence and multiplicity of positive solutions for the system of second order quasilinear boundary value problems u p 1 f t u v v q 1 g t u v u 0 u 1 0 v 0 v 1 0 where p q 1 and f g c 0 1 r 2 r r 0 based on a priori estimates achieved by utilizing jensen s inequality for nonnegative concave functions and homogeneous operators we use fixed point index theory to establish the main results

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

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

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

existence and multiplicity results for dirichlet problems with p laplacian in this paper we apply morse theory to study the existence of nontrivial solutions of p laplacian type dirichlet boundary value problems

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

on the spectrum of two different fractional operators in this paper we deal with two non local operators that are both well known and widely studied in the literature in connection with elliptic problems of fractional type more precisely for a fixed s 0 1 we consider the integral definition of the fractional laplacian given by where c n s is a positive normalizing constant and another fractional operator obtained via a spectral definition that is where e i i are the eigenfunctions and the eigenvalues of the laplace operator in with homogeneous dirichlet boundary data while a i represents the projection of u on the direction e i the aim of this paper is to compare these two operators with particular reference to their spectrum in order to emphasize their differences

infinitely many solutions for kirchhoff problems with lack of compactness abstract in this paper we consider the following kirchhoff problem 1 b r 3 u 2 d x u v x u f u x r 3 u h 1 r 3 where b is a positive constant assume that f u is an odd function of u under some appropriate assumptions on v but without radial symmetry or compactness hypotheses we establish the existence of infinitely many solutions to the above problem by using an approximation method employed by sato and shibata 2018 and some new arguments

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

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

positive solutions of nonlinear problems involving the square root of the laplacian abstract we consider nonlinear elliptic problems involving a nonlocal operator the square root of the laplacian in a bounded domain with zero dirichlet boundary conditions for positive solutions to problems with power nonlinearities we establish existence and regularity results as well as a priori estimates of gidas spruck type in addition among other results we prove a symmetry theorem of gidas ni nirenberg type

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

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

expressibility and parallel complexity it is shown that the time needed by a concurrent read concurrent write parallel random access machine cram to check if an input has a certain property is the same as the minimal depth of a first

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

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