a critical p biharmonic system with negative exponents abstract in this paper we are devoted to the study of critical p biharmonic system with a parameter which involves strongly coupled critical nonlinearities and negative exponents we prove that there exists a positive constant such that the above problem admits at least two solutions if 0

three solutions for a dirichlet boundary value problem involving the p laplacian abstract in this paper we prove the existence of at least three weak solutions for the problem p u f x u a x u p 2 u in u 0 on our main tool is a recent three critical points theorem of b ricceri on a three critical points theorem arch math basel 75 2000 220 226

separability of very noisy mixed states and implications for nmr quantum computing we give a constructive proof that all mixed states of n qubits in a sufficiently small neighborhood of the maximally mixed state are separable unentangled the construction provides an explicit representation of any such state as a mixture of product states we give upper and lower bounds on the size of the neighborhood which show that its extent decreases exponentially with the number of qubits the bounds show that no entanglement appears in the physical states at any stage of present nmr experiments though this result raises questions about nmr quantum computation further analysis would be necessary to assess the power of the general unitary transformations which are indeed implemented in these experiments in their action on separable states

asymptotic behavior and existence results for the biharmonic problems involving rellich potentials abstract in this paper we first investigate the following limit problem l u 2 u u x 4 u 2 s 1 x s u 0 x r n 0 where 0 s 4 2 s 2 n s n 4 0 1 16 n 2 n 4 2 the asymptotic properties at the origin and infinity of its ground state solutions are obtained which minimize the related best rellich sobolev constant secondly we study the following critical biharmonic problem l u u 2 s 2 u x s u q 2 u x u u n 0 x where r n n 5 is a smooth bounded domain containing the origin 0 s 4 2 q 2 2 n n 4 0 by variational argument the existence of nontrivial solutions to the problem is established

unique decompositions faces and automorphisms of separable states let sk be the set of separable states on b cm cn admitting a representation as a convex combination of k pure product states or fewer if m 1 n 1 and k max m n we show that sk admits a subset vk such that vk is dense and open in sk and such that each state in vk has a unique decomposition as a convex combination of pure product states and we describe all possible convex decompositions for a set of separable states that properly contains vk in both cases we describe the associated faces of the space of separable states which in the first case are simplexes and in the second case are direct convex sums of faces that are isomorphic to state spaces of full matrix algebras as an application of these results we characterize all affine automorphisms of the convex set of separable states and all automorphisms of the state space of b cm cn that preserve entanglement and separability

on the critical dimension of a fourth order elliptic problem with negative exponent we study the regularity of the extremal solution of the semilinear biharmonic equation 2u u 1 u 2 on a ball b rn under navier boundary conditions u u 0 on b where 0 is a parameter while 0 0 are fixed constants it is known that there exists such that for there is no solution while for 0 and it is singular supbu 1 for n 9 0 and 0 with small our proof for the singularity of extremal solutions in dimensions n 9 is based on certain improved hardy rellich inequalities

multiplicity of solutions for a fourth order problem with exponential nonlinearity abstract let b be the unit ball in r n n 5 and n be the exterior unit normal vector on the boundary we consider radial solutions to 2 u e u in b u 0 and u n 0 on b where 0 we show that there exists a unique s 0 such that if s there is a radial singular solution if 5 n 12 then for s there exist infinitely many regular radial solutions and as s the number of such solutions goes to infinity if n 13 we prove uniqueness of smooth radial solutions we derive similar results for the same equation with navier boundary conditions

polyharmonic k hessian equations in rn abstract this work is focused on the study of the nonlinear elliptic higher order equation m u s k u f x r n where the k hessian s k u is the kth elementary symmetric polynomial of eigenvalues of the hessian matrix of the solution and the datum f belongs to a suitable functional space this problem is posed in r n and we prove the existence of at least one solution by means of topological fixed point methods for suitable values of m n questions related to the regularity of the solutions and extensions of these results to the nonlocal setting are also addressed on the way to construct these proofs some technical results such as a fixed point theorem and a refinement of the critical sobolev embedding which could be of independent interest are introduced

on a class of boundary value problems involving the p biharmonic operator abstract a nonlinear boundary value problem involving the p biharmonic operator is investigated where p 1 it describes various problems in the theory of elasticity e g the shape of an elastic beam where the bending moment depends on the curvature as a power function with exponent p 1 we prove existence of solutions satisfying a quite general boundary condition that incorporates many particular boundary conditions which are frequently considered in the literature

positive radial solutions for quasilinear biharmonic equations abstract we study the quasilinear biharmonic problem u p 2 u f x u g x x b 1 u u 0 x b 1 where b 1 x r n x 1 with n 2 criteria are established for the existence of at least one positive radial solution of the problem explicit lower and upper bounds of the solutions are obtained in our results

entire solutions for a class of variational problems involving the biharmonic operator and rellich potentials abstract we study existence multiplicity and qualitative properties of entire solutions for a noncompact problem related to second order interpolation inequalities with weights more precisely we deal with the following family of equations 2 u x 4 u x u q 2 u in r n where n 5 q 2 n q n 4 2 and r is smaller than the rellich constant

some existence results of bounded variation solutions to 1 biharmonic problems abstract in this paper we establish some existence results to a fourth order quasilinear elliptic equation involving the 1 biharmonic operator formally given by 1 2 u u u we consider different geometrical assumptions in the nonlinearity and use variational methods in order to get nontrivial ground state solutions

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

asymptotics for the principal eigenvalue of the p biharmonic operator on the ball as p approaches 1 abstract using the estimates of the principal eigenvalue of the p biharmonic operator on the ball from below and from above we provide its asymptotic analysis as p 1

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

multiplicity theorems for the dirichlet problem involving the p laplacian abstract in this paper we establish some results on the existence of at least three weak solutions for dirichlet problems involving the p laplacian by a variational approach

entire solutions for a class of elliptic equations involving p biharmonic operator and rellich potentials abstract we study existence multiplicity and qualitative properties of entire solutions for a noncompact problem related to p biharmonic type equations with weights more precisely we deal with the following family of equations p 2 u x 2 p u p 2 u x u q 2 u in r n where n 2 p p 1 q p n q p n 2 p and r is smaller than the rellich constant

the regularity and stability of solutions to semilinear fourth order elliptic problems with negative exponents we examine the regularity of the extremal solution of the nonlinear eigenvalue problem on a general bounded domain in n with navier boundary condition u u on firstly we prove the extremal solution is smooth for any p 1 and n 4 which improves the result of guo and wei discrete contin dynam syst a 34 2014 2561 2580 secondly if p 3 n 3 we prove that any radial weak solution of this nonlinear eigenvalue problem is smooth in the case which completes the result of davila et al math annalen 348 2009 143 193 finally we also consider the stability of the entire solution of 2 u l u p in n with u 0

a note on nonlinear biharmonic equations with negative exponents abstract in this paper we study radially symmetric entire solutions of 2 u u q u 0 in r 3 we find the asymptotic behavior of these solutions for any q 1 and prove that for any q 3 there exists a solution with exactly linear growth

on a p biharmonic problem with no flux boundary condition the study of fourth order partial differential equations has flourished in the last years however a p biharmonic problem with no flux boundary condition has never been considered before not even for constant p this is an important step further since surfaces that are impermeable to some contaminants are appearing quite often in nature hence the significance of such boundary condition by relying on several variational arguments we obtain the existence and the multiplicity of weak solutions to our problem we point out that although we use a mountain pass type theorem in order to establish the multiplicity result we do not impose an ambrosettirabinowitz type condition nor a symmetry condition on our nonlinearity f

three solutions for a class of quasilinear elliptic systems involving the p q laplacian abstract the existence of at least three weak solutions is established for a class of quasilinear elliptic systems involving the p q laplacian with dirichlet boundary condition our technical approach is based on the three critical points theorem obtained by b ricceri

multiplicity of solutions for a fourth order equation with power type nonlinearity let b be the unit ball in mathbb r n n 3 and n be the exterior unit normal vector on the boundary we consider radial solutions to delta 2 u lambda 1 rm sign p u p quad rm in b quad u 0 quad frac partial u partial n 0 quad rm on partial b where 0 for positive p we assume 5 n 12 and p frac n 4 n 4 or n 13 and frac n 4 n 4 0 such that if s there exists a radial weakly singular solution for s there exist infinitely many regular radial solutions and the number of radial regular solutions goes to infinity as s

a new proof of i guerra s results concerning nonlinear biharmonic equations with negative exponents abstract here we give a self contained new proof of the asymptotic behavior of the radially symmetric entire solution of 2 u u p in r 3 these results were obtained by i guerra in 4 our proof is much more direct and simpler

the distillability of entanglement of bipartite reduced density matrices of a tripartite state the relation between the distillability of entanglement of three bipartite reduced density matrices from a tripartite pure state has been studied in phys rev a 84 012325 2011 we extend this result to the tripartite mixed state of rank at most three in particular we show that if the state has two bipartite reduced density operators with rank two then the third bipartite reduced density operator additionally having non positive partial transpose non ppt is distillable in contrast we show that the tripartite ppt state with two reduced density operators of rank two is a three qubit fully separable state we obtain these facts by proving a conjectured matrix inequality for the bipartite matrix m with schmidt rank at most three this is one of the main results of this paper we also prove it for some m with arbitrary schmidt rank

perturbation results of critical elliptic equations of caffarelli kohn nirenberg type abstract we find for small e positive solutions to the equation div x 2a u x 2 1 a u 1 ek x u p 1 x bp in r n which branch off from the manifold of minimizers in the class of radial functions of the corresponding caffarelli kohn nirenberg type inequality moreover our analysis highlights the symmetry breaking phenomenon in these inequalities namely the existence of non radial minimizers

some remarks on a three critical points theorem abstract some remarks on a strict minimax inequality which plays a fundamental role in ricceri s three critical points theorem are presented as a consequence some recent applications of ricceri s theorem to nonlinear boundary value problems are revisited by obtaining more precise conclusions

on polyharmonic regularizations of k hessian equations variational methods abstract this work is devoted to the study of the boundary value problem 1 u 1 k s k u f x r n u n u n 2 u n 1 u 0 x where the k hessian s k u is the k th elementary symmetric polynomial of eigenvalues of the hessian matrix and the datum f obeys suitable summability properties we prove the existence of at least two solutions of which at least one is isolated strictly by means of variational methods we look for the optimal values of n that allow the construction of such an existence and multiplicity theory and also investigate how a weaker definition of the nonlinearity permits improving these results

some singular equations modeling mems in the past fifteen years mathematical models for microelectromechanical systems mems have been the subject of several studies in particular due to the interesting qualitative properties they feature still most research is devoted to an illustrative but simplified model which is deduced from a more complex model when the aspect ratio of the device vanishes the so called vanishing or small aspect ratio model the analysis of the aforementioned complex model involving a moving boundary has started only recently and an outlook of the results obtained so far in this direction is provided in this survey

multiplicity results for solutions of p biharmonic problems abstract in this paper we deal with the multiplicity existence of solutions for the following p biharmonic equation p 2 u u p 2 u u q 2 u in u u 0 on where is a bounded domain in r n p 2 u u p 2 u p q p n p n 2 p r is a parameter when p q p we prove that the above problem possesses infinitely many solutions while when q p a multiplicity existence result is obtained

existence and multiplicity of solutions for fourth order boundary value problems with parameters abstract in this paper we study the existence and multiplicity of the solutions for the fourth order boundary value problem bvp u 4 t u t u t f t u t 0 t 1 u 0 u 1 u 0 u 1 0 where f 0 1 r r is continuous r and r are parameters by means of the idea of the decomposition of operators shown by chen w y chen a decomposition problem for operators xuebao of dongbei renmin university 1 1957 95 98 see also m krasnosel skii topological methods in the theory of nonlinear integral equations gostehizdat moscow 1956 and the critical point theory we obtain that if the pair is on the curve 2 4 satisfying 2 2 then the above bvp has at least one two three and infinitely many solutions for being in different interval respectively

existence of radial solutions to biharmonic k hessian equations abstract this work presents the construction of the existence theory of radial solutions to the elliptic equation 2 u 1 k s k u f x x b 1 0 r n provided either with dirichlet boundary conditions u n u 0 x b 1 0 or navier boundary conditions u u 0 x b 1 0 where the k hessian s k u is the k th elementary symmetric polynomial of eigenvalues of the hessian matrix and the datum f l 1 b 1 0 while r we prove the existence of a caratheodory solution to these boundary value problems that is unique in a certain neighborhood of the origin provided is small enough moreover we prove that the solvability set of is finite giving an explicity bound of the extreme value

bessel pairs and optimal hardy and hardy rellich inequalities we give necessary and sufficient conditions on a pair of positive radial functions v and w on a ball b of radius r in rn n 1 so that the following inequalities hold for all u in c 0 infty b label one int limits b v x nabla u 2 dx geq int limits b w x u 2dx label two int limits b v x delta u 2 dx geq int limits b w x nabla u 2 dx n 1 int limits b left frac v x x 2 frac v r x x right nabla u 2dx this characterization makes a very useful connection between hardy type inequalities and the oscillatory behaviour of certain ordinary differential equations and helps in the identification of a large number of such couples v w that we call bessel pairs as well as the best constants in the corresponding inequalities this allows us to improve extend and unify many results old and new about hardy and hardy rellich type inequalities such as those obtained by caffarelli et al compos math 53 259 275 1984 brezis and vazquez revista mat univ complutense madrid 10 443 469 1997 wang and willem j funct anal 203 550 568 2003 adimurthi et al proc am math soc 130 489 505 2002 and many others

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

the flow shop problem with no idle constraints a review and approximation the makespan minimization problem in flow shops with no idle constraints on machines is considered the latter means that each machine once started must process all its operations without intermediate idle time until all those operations are completed the problem is known to be strongly np hard already for three machines while being based on a geometrical approach we propose several polynomial time heuristics for the general case and for special cases of 3 and 4 machines which provide asymptotically optimal solutions for the increasing number of jobs a comprehensive review of relevant results is also presented

classification of radial solutions for semilinear elliptic systems with nonlinear gradient terms abstract we are concerned with the classification of positive radial solutions for the system u v p v f u where p 0 and f c 1 0 is a nondecreasing function such that f t 0 for all t 0 we show that in the case where the system is posed in the whole space r n such solutions exist if and only if 1 0 s f t d t p 2 p 1 d s this is the counterpart of the keller osserman condition for the case of single semilinear equation similar optimal conditions are derived in case where the system is posed in a ball of r n if f t t q q 1 using dynamical system techniques we are able to describe the behaviour of solutions at infinity in case where the system is posed in the whole r n or around the boundary in case of a ball

existence of three solutions for a quasilinear two point boundary value problem abstract in this paper we investigate a quasilinear second order differential equation with dirichlet boundary conditions the existence of an open interval of parameters which ensures this problem admits at least three solutions is determined by using the critical point theory

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 solutions for p q biharmonic systems abstract in this paper we establish the existence of at least three solutions to a navier boundary problem involving the p q biharmonic systems the technical approach is mainly based on a three critical points theorem of b ricceri

radial singular solutions for a fourth order equation with negative exponents abstract in this paper we consider the radial singular solution of 2 u 1 u p in r n with u 0 p 0 by using some elementary ordinary differential equation arguments we prove the above equation admits no radial singular solution for n 3 p 3 in addition the exact asymptotic behaviors of the radial singular solution as r 0 is established for p 1 n 4 which have a significant difference with the explicit singular solution as a product the singularity of any radial singular solution is of type ii if p 1 n 4

existence of multiple positive solutions to singular elliptic systems involving critical exponents abstract this paper is devoted to investigate multiple positive solutions to a singular elliptic system where the nonlinearity involves a combination of concave and convex terms by exploiting the effect of the coefficient of the critical nonlinearity and a variational method we establish the main result which is based on the argument of the compactness

a fourth order model for mems with clamped boundary conditions the dynamical and stationary behaviors of a fourth order equation in the unit ball with clamped boundary conditions and a singular reaction term are investigated the equation arises in the modeling of microelectromechanical systems mems and includes a positive voltage parameter lambda it is shown that there is a threshold value lambda 0 of the voltage parameter such that no radially symmetric stationary solution exists for lambda lambda while at least two such solutions exist for lambda in 0 lambda local and global well posedness results are obtained for the corresponding hyperbolic and parabolic evolution problems as well as the occurrence of finite time singularities when lambda lambda

on some fourth order semilinear elliptic problems in r n the existence of two solutions for a fourth order semilinear elliptic problem involving critical growth from the viewpoint of sobolev embedding was established the basic tools used in the analysis were the mountain pass theorem constrained minimization and concentration compactness principle the existence of positive principle eigenvalues for the corresponding linear elliptic problem was also investigated

multiple solutions for a two point boundary value problem abstract in this paper using a recent result of ricceri we prove two multiplicity theorems for the problem u f u g x u u 0 u 1 0 extending a previous result that g bonanno obtained for 0

infinitely many solutions for a perturbed nonlinear navier boundary value problem involving the p biharmonic abstract by using critical point theory we establish the existence of infinitely many weak solutions for a class of elliptic navier boundary value problems depending on two parameters and involving the p biharmonic operator

multiplicity results for perturbed fourth order kirchhoff type elliptic problems using variational methods and critical point theory we establish multiplicity results of nontrivial and nonnegative solutions for a perturbed fourth order kirchhoff type elliptic problem

optimal ensemble length of mixed separable states the optimal pure state ensemble length of a separable state a is the minimum number of pure states needed in convex combination to construct a we study the set of all separable states with optimal pure state ensemble length equal to k or fewer lower bounds on k are found below which these sets have measure 0 in the set of separable states in the bipartite case and the multiparticle case where one of the particles has significantly more quantum numbers than the rest the lower bounds are sharp a consequence of our results is that for all two particle systems except possibly those with one qubit or those with a nine dimensional hilbert space and for all systems with more than two particles the optimal pure state ensemble length for a randomly picked separable state is with probability 1 greater than the state s rank in bipartite systems with probabilty 1 it is greater than 1 4 the rank raised to the 3 2 power and in a system of p qubits with probability 1 it is greater than 22p 1 2p i e alm

generalizations of quantum discord the original definition of quantum discord of bipartite states was defined over projective measurements in this paper we discuss some generalizations of it these generalizations are defined over general measurements rank 1 general measurements or neumark extension measurements we investigate the non negativity zero discord sets of all these quantum discords and some properties of them

a boundary value problem for fourth order elastic beam equations multiplicity results for a fourth order nonlinear boundary value problem are presented the proof of the main result is based on the critical point theory

on a fourth order elliptic equation with critical nonlinearity in dimension six abstract this paper is devoted to the study of the following nonlinear elliptic problem 2 u ku 5 u 0 in u u 0 on where k is a positive function and is a bounded and smooth domain in r 6 we prove a version of morse lemma at infinity for this problem using this morse lemma at infinity we characterize the critical points at infinity of the associated functional and we give an existence result

existence of three solutions for a class of dirichlet quasilinear elliptic systems involving the p1 pn laplacian abstract in this paper we prove the existence of at least three weak solutions for the quasilinear elliptic systems p 1 u 1 f u 1 x u 1 u 2 u n 0 in p 2 u 2 f u 2 x u 1 u 2 u n 0 in p n u n f u n x u 1 u 2 u n 0 in u i 0 for 1 i n on our main tool is a recent three critical points theorem of ricceri b ricceri on a three critical points theorem arch math basel 75 2000 220 226

uniqueness theorem for integral equations and its application abstract this paper is devoted to answering a question asked recently by y li regarding geometrically interesting integral equations the main result is to give a necessary and sufficient condition on the parameters so that the integral equation with parameters to be discussed in this paper have regular solutions in the case such condition is satisfied we will write down the exact solution as its application of our method we should show that the non existence theory of the solutions of prescribed scalar curvature equation on s n can be generalized to that of prescribed branson paneitz q curvature equations on s n

symmetry properties in systems of semilinear elliptic equations abstract we investigate symmetry properties of solutions of systems of semilinear elliptic equations the two main tools we use consist of the maximum principle and the device of moving parallel planes to a critical position and then showing that the solution is symmetric about the limiting plane an application to the belousov zhabotinskii chemical reaction is given

multiple quenching solutions of a fourth order parabolic pde with a singular nonlinearity modeling a mems capacitor finite time singularity formation in a fourth order nonlinear parabolic partial differential equation pde is analyzed the pde is a variant of a ubiquitous model found in the field of microelectromechanical systems mems and is studied on a one dimensional 1d strip and the unit disc the solution itself remains continuous at the point of singularity while its higher derivatives diverge a phenomenon known as quenching for certain parameter regimes it is shown numerically that the singularity will form at multiple isolated points in the 1d strip case and along a ring of points in the radially symmetric two dimensional case the location of these touchdown points is accurately predicted by means of asymptotic expansions the solution itself is shown to converge to a stable self similar profile at the singularity point analytical calculations are verified by use of adaptive numerical methods which take advantage of symmetries exhibited by the underlying pde to accurately resolve solutions very close t

an exact estimate result for a class of singular equations with critical exponents abstract we consider the singular boundary value problem u h x u u p 1 in u 0 on with p 2 n n 2 0 1 it is well known that there exists 0 such that the problem has a solution for all 0 and no solution for we obtain an exact result for p h

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

the critical dimension for a fourth order elliptic problem with singular nonlinearity we study the regularity of the extremal solution of the semilinear biharmonic equation delta 2 u frac lambda 1 u 2 which models a simple micro electromechanical system mems device on a ball b subset mathbb r n under dirichlet boundary conditions u partial nu u 0 on partial b we complete here the results of lin and yang 14 regarding the identification of a pull in voltage 0 such that a stable classical solution u with 0 our main result asserts that the extremal solution u lambda is regular rm sup b u lambda 1 provided n leqq 8 while u lambda is singular rm sup b u lambda 1 for n geqq 9 in which case 1 c 0 x 4 3 leqq u lambda x leqq 1 x 4 3 on the unit ball where c 0 left frac lambda overline lambda right frac 1 3 and bar lambda frac 8 9 left n frac 2 3 right left n frac 8 3 right

three solutions for a navier boundary value problem involving the p biharmonic the existence of at least three weak solutions is established for a class of navier boundary value problem involving the p biharmonic our technical approach is based on the three critical points theorem obtained by b ricceri

positive solutions of semilinear biharmonic equations with critical sobolev exponents in this paper we study the critical growth biharmonic problem with a parameter and establish uniform lower bounds for which is the supremum of the set of related to the existence of positive solutions of the biharmonic problem

a travelling salesman approach to solve the f no idle cmax problem this paper investigates the f no idle cmax problem where machines work continuously without idle time intervals the idle characteristic is a very strong constraint and it affects seriously the value of cmax criterion we treat here only the permutation flow shop configuration for machine no idle problems with the objective to minimise the makespan based on the idea that this problem can be modelled as a travelling salesman problem an adaptation of the well known nearest insertion rule is proposed to solve it a computational study shows the result quality

a result on elliptic systems with neumann conditions via ricceri s three critical points theorem abstract this paper is concerned with the study of the existence of nontrivial solutions for elliptic systems involving the p laplacian by using the result of b ricceri a three critical points theorem revisited nonlinear anal 2008 in press doi 10 1016 j na 2008 04 010 we establish the existence of at least three solutions

compactness along the branch of semistable and unstable solutions for an elliptic problem with a singular nonlinearity we study the branch of semistable and unstable solutions i e those whose morse index is at most 1 of the dirichlet boundary value problem u f x 1 u 2 on a bounded domain n which models among other things a simple electrostatic microelectromechanical system mems device we extend the results of 11 relating to the minimal branch by obtaining compactness along unstable branches for 1 n 7 on any domain and for a large class of permittivity profiles f we also show the remarkable fact that powerlike profiles f x x can push back the critical dimension n 7 of this problem by establishing compactness for the semistable branch on the unit ball also for n 8 and as long as as a byproduct we are able to follow the second branch of the bifurcation diagram and prove the existence of a second solution for in a natural range in all these results the conditions on the space dimension and on the power of the profile are essentially sharp 2007 wiley periodicals inc

multiple solutions for p laplacian type equations abstract in this paper we establish the existence of three weak solutions of an equation which involves a general elliptic operator in divergence form in particular a p laplacian operator while the nonlinearity has a p 1 sublinear growth at infinity this result completes some recent papers where mountain pass type solutions were obtained providing the nonlinear term via a p 1 superlinear growth at infinity fulfilling an ambrosetti rabinowitz type condition in our case an abstract critical point result is applied proved by g bonanno g bonanno some remarks on a three critical points theorem nonlinear analysis 54 2003 651 665

vector summation in banach space and polynomial algorithms for flow shops and open shops we consider the flow shop and the open shop problems with m machines and n jobs m is the total processing time of the most loaded machine t is the maximum operation length two functions are defined m is the minimum number such that the optimum of any flow shop is at most m m t m is the minimum number for which the inequality m m t ensures that the optimum of the open shop is equal to m upper and lower bounds for functions m and m are derived and two polynomial algorithms are suggested the first one delivers an optimal schedule of an open shop provided m m t for some function m m the second one computes a near optimal approximation schedule of a flow shop both algorithms are based on a vector summation algorithm in m dimensional banach space