nodal type bound states of schrodinger equations via invariant set and minimax methods for nonlinear schrodinger equations in the entire space we present new results on invariant sets of the gradient flows of the corresponding variational functionals the structure of the invariant sets will be built into minimax procedures to construct nodal type bound state solutions of nonlinear schrodinger type equations 2004 elsevier inc all rights reserved

infinitely many nodal solutions for nonlinear elliptic equations with combined nonlinearities and lack of symmetry abstract we present a new approach to studying a parametric nonlinear dirichlet problem driven by a nonhomogeneous differential operator and with a reaction which is concave i e p 1 sublinear near zero and convex i e p 1 superlinear near the reaction term is not assumed to be odd we show that for all small values of the parameter 0 the problem has infinitely many nodal solutions

multiple solutions of nonlinear schrodinger equations via flow invariance and morse theory we prove the existence of multiple bound states of the nonlinear schrodinger equation u v x u f u here the linear potential v is continuous and bounded from below and the nonlinearity f is of asymptotically linear type we show that under certain assumptions on the spectrum of the schrodinger operator v and the asymptotic behaviour of f u u the above equation has at least four non trivial solutions two of them sign changing

a note on multiple solutions of some semilinear elliptic problems abstract we discuss the existence of multiple solutions of nonlinear elliptic equations by a combination of variational topological methods and the generalized conley index theory we obtain several positive solutions and sign changing solutions our main point is to show the usefulness of the morse inequalities for morse decompositions in the generalized conley index theory

on an asymptotically p linear p laplacian equation in rn abstract in this paper we study the existence of positive solutions and sign changing solutions for the p laplacian equation p u g x u p 2 u f x u x r n where is a positive parameter and the nonlinear term f is superlinear at zero and asymptotically p linear at infinity

a compression type mountain pass theorem in conical shells in this paper we present a compression type version of the mountain pass lemma in a conical shell with respect to two norms an application to second order ordinary differential equations is included

invariant sets of descending flow in critical point theory with applications to nonlinear differential equations abstract properties of invariant sets of descending flow defined by a pseudogradient vector field of a functional in a banach space are studied in this way several critical points can be found by constructing different invariant sets on which the functional is bounded below under suitable conditions the existence of at least four critical points of a functional is proved each critical point being in a certain invariant set the theoretical results are applied to nonlinear elliptic boundary value problems and nonlinear systems of ordinary differential equations in variant cases at least four solutions are obtained for these equations

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

multiple solutions to some degenerate quasilinear equation with variable exponents via perturbation method abstract in this paper we prove the existence of signed and sign changing solutions to some degenerate quasilinear and elliptic equation containing a gradient term and involving a variable exponent our main idea is to make use of the perturbation method together with invariant sets of descending flow the fact that the exponent is not constant plays an important role

sign changing solutions on a kind of fourth order neumann boundary value problem abstract in this paper we consider the fourth order neumann boundary value problem u 4 t 2 u t u t f t u t for all t 0 1 and subject to u 0 u 1 u 0 u 1 0 using the fixed point index and the critical group we establish the existence theorem of solutions that guarantees the problem has at least one positive solution and two sign changing solutions under certain conditions

on finding sign changing solutions abstract some parameter depending linking theorems are established which allow to produce a bounded and sign changing palais smale sequence for even functionals a parameter depending fountain theorem is obtained which provides infinitely many bounded and sign changing palais smale sequences a variant mountain pass theorem is built in cones which yields bounded positive and negative palais smale sequences the usual palais smale type compactness condition and its variants are completely not necessary for these theories more exact locations of the critical sequences can be determined the abstract results are applied to the schrodinger equation with or without critical sobolev exponents u v x u u 2 2 u f x u x r n 0 where 2 is the critical sobolev exponent the existence of multiple sign changing solutions is obtained the positive and negative solutions are also gained as by products we will also show that this schrodinger problem with jumping nonlinearity is independent of the fucik spectrum

multiple solutions for semilinear resonance elliptic problems with nonconstant coefficients at infinity in this paper we study a class of semilinear elliptic equations which are completely resonant with nonconstant coefficients at origin and at infinity by applying morse theory and minimax method we obtain the existence of multiple nontrivial solutions for the equations

numerical study of the sherrington kirkpatrick model in a magnetic field we study numerically the sherrington kirkpatrick model as a function of the magnetic field h with fixed temperature t 0 6t c we investigate the finite size scaling behavior of several quantities such as the spin glass susceptibility searching for numerical evidences of the transition on the de almeida thouless line we find strong corrections to scaling which make difficult to locate the transition point this shows in a simple case the extreme difficulties of spin glass simulations in a nonzero magnetic field next we study various sum rules consequences of stochastic stability involving overlaps between three and four replicas which appear to be numerically well satisfied and in a nontrivial way finally we present data on p q for a large lattice size n 3200 at low temperature t 0 4t c where the shape predicted by the replica symmetry breaking solution of the model for a nonzero magnetic field is visible

positive and negative solutions of a class of nonlocal problems abstract the existence of nontrivial solutions of kirchhoff type equations is an important nonlocal quasilinear problem in this paper still by using the invariant sets of descent flow we obtain positive and negative solutions of a class of nonlocal quasilinear elliptic boundary value problems as follows a b u 2 u f x u in u 0 on

sign changing critical points from linking type theorems in this paper the relationships between sign changing critical point theorems and the linking type theorems of m schechter and the saddle point theorems of p rabinowitz are established the abstract results are applied to the study of the existence of sign changing solutions for the nonlinear schrodinger equation u v x u f x u u h 1 r n where f x u is a caratheodory function problems of jumping or oscillating nonlinearities and of double resonance are considered

on the eigenvalue problem for the p laplacian operator in rn abstract in this paper we discuss the following eigenvalue problem d d u p 2 d u u p 2 u in r n where 1 p n r is an eigenvalue and u 0 is an eigenfunction the main purpose of this paper is to prove that the weak form of the eigenvalue problem r n d u p 2 d u d d x r n u p 2 u d x w 1 p r n has only the trivial solution for any p

applications of local linking to critical point theory we prove the existence of nontrivial critical points of functionals with a local linking at 0 applications to elliptic boundary value problems and to hamiltonian systems are given c 1995 academic press inc

a conjectured scenario for order parameter fluctuations in spin glasses we study order parameter fluctuations opf in disordered systems by considering the behaviour of some recently introduced parameters g gc which have proven very useful in locating phase transitions we prove that both parameters g for disconnected overlap disorder averages and gc for connected disorder averages take the respective universal values 1 3 and 13 31 in the t 0 limit for any finite volume provided the ground state is unique and there is no gap in the ground state local field distributions conditions which are met in generic spin glass models with continuous couplings and no gap at zero coupling this makes g gc ideal parameters to locate phase transitions in disordered systems much like the binder cumulant for ordered systems we check our results by exactly computing opf in a simple example of uncoupled spins in the presence of random fields and the one dimensional ising spin glass at finite temperatures we discuss under which conditions the value 1 3 for g may be recovered by conjecturing different scenarios depending on whether opf are finite or vanish in the infinite volume limit in particular we discuss replica equivalence and its natural consequence lim v g v t 1 3 when opf are finite as an example of a model where opf vanish and replica equivalence does not give information about g we study the sherrington kirkpatrick spherical spin glass model by performing numerical simulations for small sizes again we find results compatible with g 1 3 in the spin glass phase

fractional elliptic problem in exterior domains with nonlocal neumann condition abstract in this paper we consider the existence of solution for the following class of fractional elliptic problem 0 1 s u u q x u p 1 u in r n n s u x 0 in where s 0 1 n 2 s r n is a bounded set with smooth boundary s denotes the fractional laplacian operator and n s is the nonlocal operator that describes the neumann boundary condition which is given by n s u x c n s r n u x u y x y n 2 s d y x

new delay dependent stability results for neural networks with time varying delay this paper studies the problem of stability analysis for neural networks nns with a time varying delay unlike the previous works the activation functions are assumed to be neither monotonic nor differentiable nor bounded by defining a more general type of lyapunov functionals some new less conservative delay dependent stability criteria are established in terms of linear matrix inequalities lmis meanwhile the computational complexity of the newly obtained stability conditions is reduced because less variables are involved numerical examples are given to illustrate the effectiveness and the benefits of the proposed method

multiple solutions of impulsive sturm liouville boundary value problem via lower and upper solutions and variational methods abstract in this paper we prove the existence of multiple solutions for second order sturm liouville boundary value problem l u f x u x 0 1 r 1 u 0 r 2 u 0 where l u p x u q x u is a sturm liouville operator r 1 u u 0 u 0 r 2 u u 1 u 1 the technical approach is fully based on lower and upper solutions and variational methods the interesting point is that the existence of four solutions and seven solutions is given

on a boundary value problem in the half space abstract in this paper we study the existence of positive solutions and sign changing solutions for the following boundary value problem in the half space u 0 in r n 1 u n g x u f u on r n 1 where is a positive parameter and the nonlinear term f is superlinear at zero and asymptotically linear at infinity

nonlinear elliptic equations with a jumping reaction abstract we study a nonlinear nonhomogeneous dirichlet problem driven by the sum of a p laplacian and a laplacian 2 p and a jumping nonlinearity under very general conditions on the reaction and without using the fucik spectrum we show that the problem has at least three nontrivial solutions and we provide sign information for all of them our approach uses critical point theory truncation and comparison techniques and morse theory

infinitely many nodal solutions for generalized logistic equations without odd symmetry on reaction abstract we consider nonlinear parametric logistic type equations driven by a nonlinear nonhomogeneous differential operator the reaction term is a caratheodory function which is not assumed to be odd the subdiffusive equidiffusive and superdiffusive cases are all treated we prove the existence of infinitely many nodal solutions we also prove a bifurcation type result describing the dependence of the set of constant sign solutions on the parameter in the superdiffusive case our approach uses variational methods together with upper lower solutions and truncation techniques and flow invariance arguments

an elliptic problem with jumping nonlinearities abstract it is proved that the elliptic problem u f x u in u 0 on on a bounded domain r n with smooth boundary has two sign changing solutions and one positive solution if f c 1 r r satisfies i f u x 0 i 1 for some i 2 lim sup u f x u u 1 lim inf u f x u u and lim sup u f x u u where 1 2 3 are eigenvalues of with 0 dirichlet boundary condition on counting with their multiplicity

multiple nodal solutions for elliptic equations abstract we assert the locations of critical points constructed for the c 1 functional by the minimax method in terms of the order structures these results are applied to nonlinear dirichlet boundary value problems to obtain the multiplicity of nodal sign changing solutions

a variational approach to subquadratic perturbations of elliptic systems abstract we consider subquadratic perturbations of semilinear elliptic systems which are in variational form a unified approach to both the cooperative and noncooperative problems is presented and a condition of nonquadraticity at infinity is introduced to treat the resonant cases through minimax techniques

the fucik spectrum of schrodinger operator and the existence of four solutions of schrodinger equations with jumping nonlinearities abstract this paper contains the existence of four solutions of schrodinger equations with jumping nonlinearities the proof procedure is supported by a lot of new results initially a consequence is rendered as a minimax principle on h 1 r n which allows us to achieve the feasibility verification of the ps condition furthermore the constructions of minimal and maximal curves of fucik spectrum in q l see the introduction for the definition of q l warrant an intensive investigation that we encounter some thorny problems is largely due to the absence of compact embedding and the appearance of essential spectrum based on a nontrivial argument we can compute critical groups of homogeneous functional at zero if a b is free of fucik spectrum and a b q l this together with convexity and concavity offers a detailed description of the two curves by a series of sophisticated tricks additionally we present a new version of morse theory in view of the fact that classical version doesn t work directly for weak smooth functional on h 1 r n finally we prove a weak maximum principle for r n which serves as a tool to get a critical point in positive and negative cone respectively and also compute critical groups of critical points of mountain pass type with the help of above preparations we attain the ultimate aim by morse inequalities and various exact homology sequences

solutions for resonant elliptic systems with nonodd or odd nonlinearities we consider by minimax techniques a class of cooperative elliptic systems at resonance if the nonlinear term is nonodd we establish some new existence theorems about nontrivial solutions if the nonlinearity is odd we obtain existence of infinitely many solutions a topic about which very little is known

infinitely many solutions to perturbed elliptic equations abstract a new version of perturbation theory is developed which produces infinitely many sign changing critical points for uneven functionals the abstract result is applied to the following elliptic equations with a hardy potential and a perturbation from symmetry u u x 2 f x u p x u in u 0 on and u u q 2 x s u p x u in u 0 on where 0 s 2 is a smooth bounded domain of r n and p x u is not odd in u infinitely many sign changing solutions are obtained

nodal and multiple solutions of nonlinear problems involving the fractional laplacian abstract this paper is devoted to the existence of nodal and multiple solutions of nonlinear problems involving the fractional laplacian s u f x u in u 0 on where r n n 2 is a bounded smooth domain s 0 1 s stands for the fractional laplacian when f is superlinear and subcritical we prove the existence of a positive solution a negative solution and a nodal solution if f x u is odd in u we obtain an unbounded sequence of nodal solutions in addition the number of nodal domains of the nodal solutions are investigated

scaling approach to order parameter fluctuations in disordered frustrated systems we present a constructive approach to obtain information about the compactness and shape of large scale lowest excitations in disordered systems by studying order parameter fluctuations opf at low temperatures we show that the parameter g that measures opf is 1 3 at t 0 provided the ground state is unique and the probability distribution for the lowest excitations is gapless and with finite weight at zero excitation energy we then apply zero temperature scaling to describe the energy and volume spectra of the lowest large scale excitations that scale with the system size and have a weight at zero energy v 0 l with v l d a low temperature expansion reveals that opf vanish like l if 0 and remain finite for space filling lowest excitations with 0 the method can be extended to extract information about the shape and fractal surface of the large scale lowest excitations

infinitely many sign changing solutions for a schrodinger equation in rn abstract in this paper we consider a schrodinger equation u a x 1 u f u applying principle of symmetric criticality and the invariant set method under some assumptions on a and f we obtain an unbounded sequence of radial sign changing solutions for the above equation in r n when 0 large enough as n 4 or n 6 0 given using fountain theorem and the principle of symmetric criticality we prove that there exists an unbounded sequence of non radial sign changing solutions for the above equation in r n

existence theorems of multiple critical points and applications in this paper using the method of invariant sets of descending flow the multiplicity of critical points of a functional is discussed a main theorem which is called chain of rings theorem is obtained the theoretical results are applied to nonlinear elliptic boundary value problems and at least seven or nine solutions are obtained

multiplicity of sign changing solutions for kirchhoff type equations abstract we are concerned with sign changing solutions to nonlocal perturbation of the nonlinear schrodinger equation in the whole r 3 by using the method of invariant sets of descending flow we prove the existence of multiple sign changing solutions provided the perturbation is sufficiently small

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

a critical point theorem in bounded convex sets and localization of nash type equilibria of nonvariational systems abstract the localization of a critical point of minimum type of a smooth functional is obtained in a bounded convex conical set defined by a norm and a concave upper semicontinuous functional a vector version is also given in order to localize componentwise solutions of variational systems the technique is then used for the localization and multiplicity of nash type positive equilibria of nonvariational systems applications are given to periodic problems

multiple critical points of saddle geometry functionals abstract we study the multiplicity of critical points for continuously differentiable functionals on real banach spaces we prove that a functional which satisfies the assumptions of the saddle point theorem and moreover is bounded from below has at least three critical points apparently there is a global minimizer and a saddle point and we show the existence of a third critical point the idea of the proof is based on the minus gradient flow this result is closely related to the three critical points theorem of h brezis and l nirenberg which assumes a local linking finally we apply the result on the dirichlet problem for semilinear stationary pdes the analysis includes for example the existence of multiple stationary solutions of bistable or allen cahn equation and semipositone problems

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

signed and sign changing solutions of bi nonlocal fourth order elliptic problem we study a class of nonlocal fourth order elliptic problem which is seldom studied because of the presence of the biharmonic operator and binonlocal terms we are interested in the existence of signed and sign changing solutions our approach is based on variational invariant sets of descending flow and cone theory our results extend and improve some recent work we study a class of nonlocal fourth order elliptic problem which is seldom studied because of the presence of the biharmonic operator and binonlocal terms we are interested in the existence of signed and sign changing solutions our approach is based on variational invariant sets of descending flow and cone theory our results extend and improve some recent work

infinitely many sign changing solutions for kirchhoff type problems in r3 abstract in this paper we consider the following nonlinear kirchhoff type problem a b r 3 u 2 u v x u f u in r 3 u h 1 r 3 where a b 0 are constants the nonlinearity f is superlinear at infinity with subcritical growth and v is continuous and coercive for the case when f is odd in u we obtain infinitely many sign changing solutions for the above problem by using a combination of invariant sets method and the ljusternik schnirelman type minimax method to the best of our knowledge there are only few existence results for this problem it is worth mentioning that the nonlinear term may not be 4 superlinear at infinity in particular it includes the power type nonlinearity u p 2 u with p 2 4

sign changing solutions for p biharmonic equations with hardy potential in the half space abstract in this paper by using the method of invariant sets of descending flow we obtain the existence of positive solutions and sign changing solutions of p biharmonic equations with hardy potential in the half space

on sign changing solution for a fourth order asymptotically linear elliptic problem in this paper we deal with a fourth order elliptic problem whose nonlinear term is asymptotically linear at both zero and infinity by using the variational method we obtain an existence result of sign changing solutions as well as positive and negative solutions

existence of solutions for some fourth order nonlinear elliptic problems in this paper we consider the existence of positive negative and sign changing solutions for some fourth order semilinear elliptic boundary value problems we present new results on invariant sets of the gradient flows of the corresponding variational functionals the structure of the invariant sets will be built into minimax procedures to construct the sign changing solutions

sign changing solutions for resonant neumann problems we consider semilinear neumann problems at resonance we are concerned with two distinct cases in the first one the potential function is unbounded and indefinite in the second case the potential function is bounded and the reaction exhibits a concave parametric term near the origin we prove two existence theorems producing sign changing solutions the first main result is a complement of the results obtained by papageorgiou and r dulescu 2015 22 the second theorem gives an answer to the open question raised by candito et al 2016 4 our approach uses variational methods together with flow invariance arguments

compactness results for schrodinger equations with asymptotically linear terms abstract we study the nonlinear problem u v x f x u x r n lim x u x 0 where the schrodinger operator v is semibounded and the nonlinearity f is linearly bounded we establish compactness of palais smale sequences and cerami sequences for the associated energy functional under general spectral theoretic assumptions applying these results we obtain existence of three nontrivial solutions if the energy functional has a mountain pass geometry

sign changing and multiple solutions of the sturm liouville boundary value problem via invariant sets of descending flow abstract in this paper we prove the existence of sign changing and multiple solutions for the second order sturm liouville boundary value problem l u f x u x 0 1 r 1 u 0 r 2 u 0 where l u p x u q x u is the sturm liouville operator r 1 u u 0 u 0 and r 2 u u 1 u 1 the technical approach is fully based on minimax methods and invariant sets of descending flow

critical point theory for indefinite functionals with symmetries letxbe a hilbert space and c1 x r be strongly indefinite assume in addition that a compact lie groupgacts orthogonally onxand that is invariant in order to find critical points of we develop the limit relative category of fournieret al in the equivariant context we use this to prove two generalizations of the symmetric mountain pass theorem and a linking theorem in the case of the mountain pass theorem the mountain range is allowed to lie in a subspace of infinite codimension also other conditions of the classical symmetric mountain pass theorem forg z 2 due to ambrosetti and rabinowitz can be weakened considerably for example we are able to deal with infinite dimensional fixed point spaces the proofs consist of a direct reduction to a relative borsuk ulam type theorem this provides a new proof even for the classical mountain pass theorem the abstract critical point theorems are applied to an elliptic system with dirichlet boundary conditions we only need a weak version of the usual superquadraticity condition the linking theorem can be applied to asymptotically linear hamiltonian systems which are symmetric with respect to a generalized symplectic group action

multiple solutions of some fourth order boundary value problems abstract for some fourth order boundary value problems several new existence theorems on multiple positive negative and sign changing solutions are obtained the critical point theory and the supersolution and subsolution method are employed to discuss this problem

sign changing saddle point abstract in this paper the saddle point theorems are generalized to a new version by showing that there exists a sign changing saddle point besides zero the abstract result is applied to the semilinear elliptic boundary value problem u f x u in u 0 on and the schrodinger equation u v x u f x u x r n u x 0 as x where r n is a bounded domain with smooth boundary the schrodinger operator v has both eigenvalues and essential spectrum the asymptotically linear case is considered which permits double resonance to be happened some existence results of sign changing solutions are established

on the existence of signed and sign changing solutions for a class of superlinear schrodinger equations this paper deals with a semilinear schrodinger equation whose nonlinear term involves a positive parameter and a real function f u which satisfies a superlinear growth condition just in a neighborhood of zero by proving an a priori estimate for a suitable class of solutions we are able to avoid further restrictions on the behavior of f u at infinity in order to prove for sufficiently large the existence of one sign and sign changing solutions minimax methods are employed to establish this result