breaking of ergodicity in expanding systems of globally coupled piecewise affine circle maps to identify and to explain coupling induced phase transitions in coupled map lattices cml has been a lingering enigma for about two decades in numerical simulations this phenomenon has always been observed preceded by a lowering of the lyapunov dimension suggesting that the transition might require changes of linear stability yet recent proofs of co existence of several phases in specially designed models work in the expanding regime where all lyapunov exponents remain positive in this paper we consider a family of cml composed by piecewise expanding individual map global interaction and finite number n of sites in the weak coupling regime where the cml is uniformly expanding we show mathematically for n 3 and numerically for n ge 3 that a transition in the asymptotic dynamics occurs as the coupling strength increases the transition breaks the milnor attractor into several chaotic pieces of positive lebesgue measure with distinct empiric averages it goes along with various symmetry breaking quantified by means of magnetization type characteristics despite that it only addresses finite dimensional systems to some extend this result reconciles the previous ones as it shows that loss of ergodicity symmetry breaking can occur in basic cml independently of any decay in the lyapunov dimension

multiple solutions for an eigenvalue problem involving p laplacian type operators abstract in this paper we establish the existence of two nontrivial weak solutions of some eigenvalue problems involving a general elliptic operator in divergence form first we study a one parameter problem under homogeneous dirichlet boundary conditions in bounded domains and then a two parameters problem under inhomogeneous nonlinear robin boundary conditions in regular bounded domains these results complete in several directions some recent papers where a ricceri type critical point theorem was used the main argument of this paper is based on two recent theorems due to arcoya and carmona 2007 in 2 which we use in a slightly modified version see section 2

three symmetric solutions for a class of elliptic equations involving the p laplacian with discontinuous nonlinearities in rn abstract using a three critical points theorem for a non differentiable functional and a sobolev embedding result we establish the existence of three radially symmetric solutions for a class of elliptic equations involving the p laplacian with discontinuous nonlinearities in the whole space

a minimax inequality and its applications to ordinary differential equations the aim of this paper is to investigate the minimax inequality which plays a fundamental role in the critical points theorem of b ricceri below equivalent formulations are shown and characterization is proved in particular for a special class of functionals as an application a multiplicity result for an ordinary dirichlet problem is emphasized

multiplicity results for a differential inclusion problem with non standard growth abstract in this paper we examine the multiplicity of solutions of a differential inclusion problem involving p x laplacian of the type p p x u v x u p x 2 u j 1 x u x j 2 x u x in u n 0 on by using the nonsmooth version of ricceri variational principle we get three critical points of the corresponding energy motreanu panagiotopoulos type functional which are the solutions of p

a three critical points theorem revisited abstract in this paper the following result is proved let x be a reflexive real banach space i r an interval x r a sequentially weakly lower semicontinuous c 1 functional bounded on each bounded subset of x whose derivative admits a continuous inverse on x j x r a c 1 functional with compact derivative assume that lim x x j x for all i and that there exists r such that sup i inf x x x j x inf x x sup i x j x then there exist a non empty open set a i and a positive real number r with the following property for every a and every c 1 functional x r with compact derivative there exists 0 such that for each 0 the equation x j x x 0 has at least three solutions in x whose norms are less than r

positive solutions of variational inequalities a degree theoretic approach if the mapping a is strictly monotone and coercive is equivalent to a fixed point equation u m u in k cf section 1 suppose that m is continuous and maps bounded subsets of k into compact ones then due to the fact that k is a retract of x it is possible to define a fixed point index of m with properties similar to those associated with the leray schauder degree 1 sect 111 in section 1 we introduce another index which is based on the previous one but has the advantage of being directly related to inequality rather than to the fixed point equation u m u in section 2 we apply this new index in order to show that if c is a closed convex subset of k then under suitable assumptions upon f the set of solutions of the inequality

existence and uniqueness results for discrete second order periodic boundary value problems abstract we prove existence and uniqueness results for solutions of second order nonlinear difference equations on a finite discrete segment with periodic boundary conditions the results are based on the notion of upper and lower solutions

multiple positive solutions of singular and nonsingular discrete problems via variational methods we employ the critical point theory to establish the existence of multiple solutions of some regular as well as singular discrete boundary value problems 2004 elsevier ltd all rights reserved msc 34b16 39a10

variational methods for non differentiable functionals and their applications to partial differential equations the interest in extending the theory of nonlinear partial differential equations pde to pde with discontinuous nonlinearities dnde has been increasing daily the reasons are twofold 1 many free boundary problems may be reduced to boundary value problems of dnde cf chang 16 71 chang and jiang 9 and fleishman and mahar 121 2 sometimes in dealing with a pde problem it is beneficial to put the original pde into a large category for instance dnde cf chang 7 simply speaking a dnde is as follows

existence of multiple solutions for second order discrete boundary value problems we give conditions on f involving pairs of discrete lower and discrete upper solutions which lead to the existence of at least three solutions of the discrete two point boundary value problem yk 1 2yk yk 1 f k yk vk 0 for k 1 n 1 y0 0 yn where f is continuous and vk yk yk 1 for k 1 n in the special case f k t p f t greater than or equal to 0 we give growth conditions on f and apply our general result to show the existence of three positive solutions we give an example showing this latter result is sharp our results extend those of avery and peterson and are in the spirit of our results for the continuous analogue c 2002 elsevier science ltd all rights reserved

an obstacle problem for nonlinear hemivariational inequalities at resonance in this paper we examine an obstacle problem for a nonlinear hemivariational inequality at resonance driven by the p laplacian using a variational approach based on the nonsmooth critical point theory for locally lipschitz functionals defined on a closed convex set we prove two existence theorems in the second theorem we have a pointwise interpretation of the obstacle problem assuming in addition that the obstacle is also a kind of lower solution for the nonlinear elliptic differential inclusion

multiplicity results for hemivariational inequalities driven by nonlocal elliptic operators abstract in this paper we extend a recently established three critical point theorem of arcoya and carmona 2007 7 relating to some functionals which are only differentiable along some directions to motreanu panagiotopoulos functionals as its applications we establish the existence of two nontrivial solutions to hemivariational inequalities driven by non local integrodifferential operators

three anti periodic solutions for second order impulsive differential inclusions via nonsmooth critical point theory abstract a nonsmooth version of a three critical point theorem of ricceri due to iannizzotto is used to obtain three anti periodic solutions for a second order impulsive differential inclusions with a perturbed nonlinearity and two parameters

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

nontrivial solutions of boundary value problems of second order difference equations we employ the critical point theory to establish the existence of nontrivial solutions for some boundary value problems of second order difference equations

multiple nontrivial solutions for some fourth order semilinear elliptic problems abstract in this paper we consider the existence of multiple nontrivial solutions for some fourth order semilinear elliptic boundary value problems the weak solutions are sought by means of morse theory and local linking

existence of three nontrivial solutions for nonlinear neumann hemivariational inequalities abstract in this paper we consider a nonlinear neumann problem driven by the p laplacian with a nonsmooth potential hemivariational inequality using minimax methods based on the nonsmooth critical point theory together with suitable truncation techniques we show that the problem has at least three nontrivial smooth solutions two of these solutions have constant sign one is positive the other negative

existence of nontrivial solutions for discrete nonlinear two point boundary value problems in this paper by using strongly monotone operator principle and critical point theory the existence of nontrivial solutions for discrete nonlinear two point boundary value problems is established two illustrative examples are given in the last part of the paper

existence of solutions for some discrete boundary value problems with a parameter in this paper the existence and multiplicity results of solutions are obtained for the discrete nonlinear two point boundary value problem bvp email protected 2u k 1 email protected k u k email protected z 1 t u 0 email protected t where t is a positive integer z 1 t 1 2 t d is the forward difference operator defined by du k u k 1 u k and f z 1 t xr r is continuous email protected r is a parameter by using the critical point theory and morse theory we obtain that the above bvp has solutions for l being in some different intervals

existence and multiplicity results for hemivariational inequalities with two parameters a multiplicity result of ricceri based on a minimax inequality is applied to locally lipschitz functionals and the existence of one or two non trivial solutions for a class of two parameter hemivariational inequalities is proved an application is given to elliptic problems with discontinuous nonlinearities on unbounded domains

multiplicity result for a discrete eigenvalue problem with discontinuous nonlinearities using a three critical points theorem for nondifferentiable functionals we investigate a class of second order difference equation with discontinuous nonlinearities a new multiplicity result is obtained

nonsmooth critical point theory on closed convex sets and nonlinear hemivariational inequalities abstract in this paper we develop a critical point theory for nonsmooth locally lipschitz functionals defined on a closed convex set extending this way the work of struwe variational methods springer berlin 1990 through a deformation result we obtain minimax principles producing critical points then we use the theory to obtain positive and negative solutions of nonlinear and semilinear hemivariational inequalities in this context we improve a result on positive solutions for semilinear elliptic problems due to nirenberg variational methods in nonlinear problems in topics in calculus of variations lecture notes in mathematics vol 1365 springer berlin 1987

three critical points for perturbed nonsmooth functionals and applications abstract a recently established three critical points theorem of ricceri relating to continuously g teaux differentiable functionals based on a minimax inequality and on a truncation argument is extended to motreanu panagiotopoulos functionals as an application multiplicity results also yielding a uniform bound on the norms of solutions are obtained for variational hemivariational inequalities depending on two parameters

a further refinement of a three critical points theorem abstract in this paper we complete the refinement process made by ricceri 2009 4 of a result established by ricceri 2000 1 which is one of the most applied abstract multiplicity theorems in the past decade a sample of application of our new result is as follows let r n n 3 be a bounded domain with smooth boundary and let 1 p q n 2 n 2 then for each 0 small enough there exists 0 such that for every compact interval a b 0 there exists 0 with the following property for every a b and every continuous function h r r satisfying lim sup h s for some s 0 n 2 n 2 there exists 0 such that for each 0 the problem u u p 1 u u q 1 u h u in u 0 on has at least three weak solutions whose norms in h 0 1 are less than