on a liouville type equation with sign changing weight in this paper we study the existence nonexistence and multiplicity of non negative solutions for the family of problems u a x e f x u u h 0 where is a bounded domain in r2 and 0 is a parameter the coefficient a x is allowed to change sign the techniques used in the proofs are a combination of upper and lower solutions the trudingermoser inequality and variational methods note that when f x u 0 the equation is of liouville type

elliptic problems with nonlinearities indefinite in sign abstract let r n n 3 be a bounded open set with smooth boundary and r we study the dirichlet problem formula with 1 q p and h k l nonnegative functions we prove existence nonexistence and multiplicity results depending on and according to the integrability properties of the ratio k p 1 h q 1

multiple positive solutions for semilinear elliptic systems with nonlinear boundary condition abstract in this paper we study the nonlinear boundary value problem u u f u u v in v v f v u v in u n g u u v v n g v u v on where is a bounded domain in r n with smooth boundary and the pairs of f u f v and g u g v are subcritical nonlinearities with the help of the nehari manifold we show existence of at least two positive solutions when one of the pairs of f u f v and g u g v is sublinear and the other one superlinear

mesh partitioning a multilevel balancing and refinement algorithm multilevel algorithms are a successful class of optimization techniques which addresses the mesh partitioning problem they usually combine a graph contraction algorithm together with a local optimization method which refines the partition at each graph level in this paper we present an enhancement of the technique which uses imbalance to achieve higher quality partitions we also present a formulation of the kernighan lin partition optimization algorithm which incorporates load balancing the resulting algorithm is tested against a different but related state of the art partitioner and shown to provide improved results

local superlinearity and sublinearity for the p laplacian we study the existence nonexistence and multiplicity of positive solutions for a family of problems updelta p u f lambda x u u in w 0 1 p upomega where is a bounded domain in mathbb r n n p and 0 is a parameter the family we consider includes the well known nonlinearities of ambrosetti brezis cerami type in a more general form namely lambda a x u q b x u r where 0 leqslant q p 1 r leqslant p 1 here the coefficient a x is assumed to be nonnegative but b x is allowed to change sign even in the critical case preliminary results of independent interest include the extension to the p laplacian context of the brezis nirenberg result on local minimization in w 0 1 p text and c 0 1 a c 1 alpha estimate for equations of the form updelta p u h x u with h of critical growth a strong comparison result for the p laplacian and a variational approach to the method of upper lower solutions for the p laplacian

parabolic monge ampere equations on riemannian manifolds on a compact riemannian manifold m g we consider the parabolic monge ampere equation t x t logdet g x hess x t detg x x t f x x 0 phiv 0 x here is a real parameter andf 0 m r are smooth functions we show existence of for all timestindependent of if 0 then t t converges exponentially towards a solution of the stationary problem ast in the special case 0 f 0 one has 0 and we determine the convergence rate t in thel2 norm more precisely

nonlinear oblique boundary value problems for nonlinear elliptic equations on considere le probleme aux valeurs limites a derivee oblique non lineaire pour des equations uniformement elliptiques quasi lineaires et totalement non lineaires d ordre 2 les operateurs elliptiques satisfont des conditions de structure naturelle on donne des theoremes d existence pour les solutions

regularity of generalized solutions of monge ampere equations on etudie la regularite interieure des solutions generalisees convexes des equations de monge ampere de la forme detd 2 d f x u du dans ou est un domaine de r n f est une fonction positive suffisamment lisse sur r r n

radially symmetric boundary value problems for real and complex elliptic monge ampere equations abstract radially symmetric dirichlet and neumann problems for real and complex monge ampere equations are considered existence of radially symmetric solutions is proved by transforming the differential equations into integral ones solvable by means of fixed point arguments then taking advantage of integral formulae regularity and convexity of the radial solutions are checked fairly weak assumptions are required in that process in the real case a priori radial symmetry is also discussed

existence of the global solution for the parabolic monge ampere equations on compact riemannian manifolds abstract on a compact riemannian manifold m g we consider the existence and nonexistence of global solutions for the parabolic monge ampere equation t log det g hess det g p f x x 0 0 x here p 1 and are real parameters f 0 m 0 are smooth functions on m if 0 then the solution of exists for all times t and t t converges exponentially towards a solution of its stationary equation as t in the case of 0 it does not have the global solution of thus we obtain the nonexistence of the positive solution for the stationary equation of

multiple positive solutions for quasilinear problems with indefinite sublinear nonlinearity abstract we have established multiplicity of nontrivial solutions for the quasilinear elliptic problem p u h x u 1 g x u in u 0 in u 0 on where r n is a smooth bounded domain p 1 1 p h is a measurable function and g r r is a caratheodory function such that g x 0 0 which is asymptotically linear we also prove existence and nonexistence results when g x t k x t p 1

nonnegative solutions of elliptic problems with sublinear indefinite nonlinearity abstract we are concerned with existence nonexistence and multiplicity of nonnegative solutions for the elliptic problem u a x u q b x u p in u 0 on where is a bounded domain in r n r 0 q 1 p 2 1 and a b are bounded functions with b x 0 and a x changes its sign

positivity results for indefinite sublinear elliptic problems via a continuity argument abstract we establish a positivity property for a class of semilinear elliptic problems involving indefinite sublinear nonlinearities namely we show that any nontrivial nonnegative solution is positive for a class of problems the strong maximum principle does not apply to our approach is based on a continuity argument combined with variational techniques the sub and supersolutions method and some a priori bounds both dirichlet and neumann homogeneous boundary conditions are considered as a byproduct we deduce some existence and uniqueness results finally as an application we derive some positivity results for indefinite concave convex type problems

a priori estimates and a liouville theorem for complex monge ampere equations on considere des solutions plurisousharmoniques u 2 des equations complexes de monge ampere definies sur un sous ensemble ouvert cc u u 2

how good is recursive bisection the most commonly used p way partitioning method is recursive bisection rb it first divides a graph or a mesh into two equal sized pieces by a good bisection algorithm and then recursively divides the two pieces ideally we would like to use an optimal bisection algorithm because the optimal bisection problem that partitions a graph into two equal sized subgraphs to minimize the number of edges cut is np complete practical rb algorithms use more efficient heuristics in place of an optimal bisection algorithm most such heuristics are designed to find the best possible bisection within allowed time we show that the rb method even when an optimal bisection algorithm is assumed may produce a p way partition that is very far way from the optimal one our negative result is complemented by two positive ones first we show that for some important classes of graphs that occur in practical applications such as well shaped finite element and finite difference meshes rb is within a constant factor of the optimal one almost always second we show that if the balance condition is relaxed so that each block in the p way partition is bounded by 2n p where n is the number of vertices of the graph then a modified rb finds an approximately balanced p way partition whose cost is within an o log p factor of the cost of the optimal p way partition

compactly supported solutions to stationary degenerate diffusion equations we consider non negative solutions of the semilinear elliptic equation in rn with n 3 u a x uq b x up where 0 q a x sign changing a a a and b x 0 is non positive under appropriate growth assumption on a at infinity we prove that all solutions in d1 2 rn are compactly supported and their support is contained in a large ball with radius determined by a when 0 x rn a x 0 has several compact connected components we give conditions under which there may or may not exist solutions which vanish identically on one or more of the components of 0 for instance we introduce a positive parameter and replace a by a a we then show that for small all solutions have compact support and there exist solutions with supports in any combination of these connected components of 0 for large and p 1 the solution is unique and supported in all of 0 we also prove the existence of the limit of this solution which solves w a wq and lim x w x 0 the analysis is based on comparison arguments and a priori bounds

a free boundary problem involving convection and singular absorption abstract we study nonnegative stationary solutions of a parabolic equation with convection and singular absorption the convection term depends on a real parameter existence and qualitative properties of free boundary solutions are established for r

the dirichlet problem for nonlinear second order elliptic equations i monge ampegre equation on considere le probleme de dirichlet pour des equations elliptiques non lineaires pour une fonction reelle u definie dans la fermeture d un domaine borne dans r n avec une frontiere c

multiple solutions for a semilinear problem with combined terms and nonlinear boundary conditions abstract we consider the problem u u f x u in u h x u q 2 u on where r n is a smooth bounded domain n 3 1 q 2 and h belongs to an appropriated lebesgue space in our main results we suppose that f is an asymptotically linear function and we obtain multiplicity of solutions when the norm of h is small we also present a multiplicity result in the case that f is nonquadratic at infinity

positive solutions of a one dimensional indefinite capillarity type problem a variational approach abstract we prove the existence and the multiplicity of positive solutions of the one dimensional capillarity type problem u 1 u 2 a x f u u 0 0 u 1 0 where a l 1 0 1 changes sign and f 0 0 is continuous and has a power like behavior at the origin and at infinity our approach is variational and relies on a regularization procedure that yields bounded variation solutions which are of class w loc 2 1 and hence satisfy the equation pointwise almost everywhere on each open interval where the weight function a has a constant sign

global existence of solutions to parabolic monge ampere equations on riemannian manifolds abstract in this paper we study the cauchy problem of the parabolic monge ampere equation u t log g 1 det g i j i j u n log u in m n 0 u x 0 u 0 x in m n where m n is a compact complete riemannian manifold of dimension n 2 g i j denotes the metric of m n g det g i j 0 and u 0 x 1 is a smooth function we prove the global existence of solutions

equations de monge ampere reelles resume le probleme de dirichlet pour l equation de monge ampere det ij exp x u o u est un ouvert borne strictement convexe de r n est etudie par la methode de continuite la regularite sur de la solution est obtenue pour n 2 l hypothese 0 n est pas requise un probleme variationnel liea la fonctionnelle i det ij dx pour convexe et nulle sur est resolu

infinite solutions for a class of brezis nirenberg equations with an indefinite linear and nonlinear terms in sign let p q 0 1 in the present paper we study the existence of infinitely many nontrivial solutions for a class of brezis nirenberg equation du v x u a x u p 1u b x u q 1u inr n where n 3 a x and b x have a contrary sign and satisfy suitable conditions the proof is based on the variant fountain theorem established by zou

singular boundary value problems for the monge ampere equation abstract given a strictly convex smooth and bounded domain in r n we consider solving the monge ampere equation det d 2 u f x u for solutions in c c with zero boundary value where the nonlinearity f x t could be singular at t 0 we will show that under some fairly general assumptions on f the above dirichlet problem admits a negative convex solution in uniqueness of such solutions is then established for a wide class of nonlinearities f x t as a consequence of a comparison principle

the nehari manifold for a semilinear elliptic system involving sign changing weight functions abstract in this paper we study the combined effect of concave and convex nonlinearities on the number of solutions for a semilinear elliptic system e with sign changing weight function with the help of the nehari manifold we prove that the system has at least two nontrivial nonnegative solutions when the pair of the parameters belongs to a certain subset of r 2

stabilization of solutions of a degenerate nonlinear diffusion problem i 4 u f u in l l x r u tl t o in r u x 0 uo x in l l where m 1 is a parameter f is locally lipschitz continuous f 0 0 and u is bounded problems of this form arise in a number of areas of science for instance in models for gas or fluid flow in porous media 2 and for the spread of certain biological populations 13 161 this paper is divided into two parts in part i we consider what may be called the motivating example problem i which consists of problem i with the special choice f u u 1 u u a 1 for suitably restricted parameters a we begin by describing in detail the set l of nonnegative equilibrium solutions of problem i clearly 8 l contains the trivial solution u 0 for all l 0 write 8 l e l o in the description of z l there are two critical parameter values lo and l1 with 0 lo l1 j we show that i g l 4 for 0 l lo

indefinite weight nonlinear problems with neumann boundary conditions abstract we present a multiplicity result of positive solutions for the neumann problem associated with a second order nonlinear differential equation of the following form u a t g u 0 where the weight function a t has indefinite sign the only assumption we make for the nonlinear term g u is that its primitive g u presents some oscillations at infinity expressed by the condition involving lim g u u 2 0 lim g u u 2 as an application we obtain multiple radially symmetric solutions for neumann problems associated with u a x g u 0

regularity estimates of solutions to complex monge ampere equations on hermitian manifolds in this paper we obtain the bedford taylor interior c2 estimate and local calabi c3 estimate for the solutions to complex monge ampere equations on hermitian manifolds

regularity of solutions of two dimensional monge ampere equations in the paper we investigate the regularity of solutions z x y 6 c1 1 fl resp c1 1 fl of elliptic monge ampere equations of the form ar 2bs ct r rt s2 e it is shown that z x y c2 t fi resp c2 a 0 with corresponding a priori estimates if a b c e 6 ca q x r3 the results are deduced via the cam panato technique for equations of variational structure invoking a legendre like transformation

multiple positive solutions to elliptic boundary blow up problems abstract we prove the existence of multiple positive radial solutions to the sign indefinite elliptic boundary blow up problem u a x a x g u 0 x 1 u x x 1 where g is a function superlinear at zero and at infinity a and a are the positive negative part respectively of a sign changing function a and 0 is a large parameter in particular we show how the number of solutions is affected by the nodal behavior of the weight function a the proof is based on a careful shooting type argument for the equivalent singular ode problem as a further application of this technique the existence of multiple positive radial homoclinic solutions to u a x a x g u 0 x r n is also considered

existence and uniqueness of solutions of nonlinear neumann problems soit d r n un domaine connexe borne lisse et on suppose qu en chaque point x d la normale exterieure n est bien definie soit a une fonction arbitraire continue de holder dans d on etudie les solutions positives du probleme de neumann non lineaire u a x g u 0 dans d u n 0 sur d ou g r r est une fonction de classe c r c 1 r 0 1

a liouville theorem on the pde det f_ i bar jmath 1 det fiȷ 1 let f be a smooth plurisubharmonic function which solves begin aligned det f i bar jmath 1 text on mathbb c n end aligned suppose that the metric omega f sqrt 1 f i bar jmath dz i wedge d bar z j is complete and f satisfies the growth condition begin aligned mathsf n 0 1 1 z 2 le f le mathsf n 0 1 z 2 text as z rightarrow infty end aligned for some mathsf n 0 0 then f is a quadratic polynomial

multiple positive solutions for dirichlet problems involving concave and convex nonlinearities abstract in this paper we study the multiplicity of positive solutions for the following dirichlet problems involving concave and convex nonlinearities u u f x u q 2 u h x u p 2 u in a u h 0 1 a where 1 q 2 p 2 2 2 n n 2 if n 3 2 if n 2 0 a r is an infinite strip in r n is a bounded domain in r n 1 and f x h x satisfy suitable conditions

elliptic problems involving the fractional laplacian in rn abstract we study the existence and multiplicity of solutions for elliptic equations in r n driven by a non local integro differential operator which main prototype is the fractional laplacian the model under consideration denoted by p depends on a real parameter and involves two superlinear nonlinearities one of which could be critical or even supercritical the main theorem of the paper establishes the existence of three critical values of which divide the real line in different intervals where p admits no solutions at least one nontrivial non negative entire solution and two nontrivial non negative entire solutions

positive periodic solutions to an indefinite minkowski curvature equation abstract we investigate the existence non existence multiplicity of positive periodic solutions both harmonic i e t periodic and subharmonic i e kt periodic for some integer k 2 to the equation u 1 u 2 a t g u 0 where 0 is a parameter a t is a t periodic sign changing weight function and g 0 0 is a continuous function having superlinear growth at zero in particular we prove that for both g u u p with p 1 and g u u p 1 u p q with 0 q 1 p the equation has no positive t periodic solutions for close to zero and two positive t periodic solutions a small one and a large one for large enough moreover in both cases the small t periodic solution is surrounded by a family of positive subharmonic solutions with arbitrarily large minimal period the proof of the existence of t periodic solutions relies on a recent extension of mawhin s coincidence degree theory for locally compact operators in product of banach spaces while subharmonic solutions are found by an application of the poincare birkhoff fixed point theorem after a careful asymptotic analysis of the t periodic solutions for

an extension to a theorem of jorgens calabi and pogorelov 1 1 det d2u 1 in rn must be a quadratic polynomial for n 2 a classical solution is either convex or concave the result holds without the convexity hypothesis a simpler and more analytical proof along the lines of affine geometry of the theorem was later given by cheng and yau 9 the first author extended the result for classical solutions to viscosity solutions 4 it was proven by trudinger and wang in 19 that the only open convex subset of rn which admits a convex c2 solution of det d2u 1 in with limx u x is rn in this paper we give the following extension to the theorem of jorgens calabi and pogorelov let u be a convex viscosity solution of det d2u 1 outside a bounded subset of

equations du type monge ampere sur les varietes riemanniennes compactes i let vn g be a c compact riemannian manifold for a suitable function on vn let us consider the change of metric g g hess and the function as a ratio of two determinants m g g 1 using the method of continuity we first solve in c the problem log m 0 c then under weak hypothesis on f we solve the general equation log m f p f in c vn using a method of iteration our study gives rise to an interesting a priori estimate on which does not occur in the complex case this estimate should enable us to solve the equation above when 0 providing we can overcome difficulties related to the invertibility of the linearised operator this open question will be treated in our next article

combined effects of concave and convex nonlinearities in some elliptic problems abstract this paper deals with a class of semilinear elliptic dirichlet boundary value problems where the combined effects of a sublinear and a superlinear term allow us to establish some existence and multiplicity results

sur les equations de monge ampere in this paper we study the real monge arapere equations det d2u f x in 0 u convex in 0 u 0 on 0 and we introduce a new method for solving these equations which enables us to show the existence of regular solutions this method uses only p d e techniques and does not use any geometrical results furthermore it enables us to solve quasilinear monge ampere equations

multiple positive solutions for a class of concave convex elliptic problems in rn involving sign changing weight in this paper we study the multiplicity of positive solutions for the following concave convex elliptic equation u u f x uq 1 g x up 1in rn u 0in rn u h1 rn where 1 q 2 p 2 2 2nn 2 if n 3 2 if n 1 2 and the parameters 0 we assume that f x f x f x is sign changing and g x a x b x where the functions f a and b satisfy suitable conditions

periodic solutions for a second order differential equation with indefinite weak singularity as a consequence of the main result of this paper efficient conditions guaranteeing the existence of a t periodic solution to the second order differential equation are established here h l t is a piecewise constant sign changing function and the non linear term presents a weak singularity at 0 i e 0 1

four positive solutions for a semilinear elliptic equation involving concave and convex nonlinearities abstract in this paper we study the combined effect of concave and convex nonlinearities on the number of positive solutions for a semilinear elliptic equation with the help of the nehari manifold and the center mass function we prove that there are at least four positive solutions for a semilinear elliptic equation in a finite strip with a hole

a c2 α estimate of the complex monge ampere equation abstract in this paper we prove a c 2 estimate for the solution to the complex monge ampere equation det u i j f with 0 f c under the assumption that u c 1 for some 1 which depends on n and

bifurcation of positive solutions for a one dimensional indefinite quasilinear neumann problem abstract we study the structure of the set of the positive regular solutions of the one dimensional quasilinear neumann problem involving the curvature operator u 1 u 2 a x f u u 0 0 u 1 0 here r is a parameter a l 1 0 1 changes sign and f c r we focus on the case where the slope of f at 0 f 0 is finite and non zero and the potential of f is superlinear at infinity but also the two limiting cases where f 0 0 or f 0 are discussed we investigate in some special configurations the possible development of singularities and the corresponding appearance in this problem of bounded variation solutions

pairs of positive periodic solutions of second order nonlinear equations with indefinite weight abstract we study the problem of the existence and multiplicity of positive periodic solutions to the scalar ode u a t g u 0 0 where g x is a positive function on r superlinear at zero and sublinear at infinity and a t is a t periodic and sign indefinite weight with negative mean value we first show the nonexistence of solutions for some classes of nonlinearities g x when is small then using critical point theory we prove the existence of at least two positive t periodic solutions for large some examples are also provided

on elliptic problems involving critical hardy sobolev exponents and sign changing function abstract in this paper we deal with the existence and nonexistence of nonnegative nontrivial weak solutions for a class of degenerate quasilinear elliptic problems with weights and nonlinearity involving the critical hardy sobolev exponent and a sign changing function some existence results are obtained by splitting the nehari manifold and by exploring some properties of the best hardy sobolev constant together with an approach developed by brezis and nirenberg

variational elliptic problems which are nonquadratic at infinity p co if n 1 2 then it is well known that the weak solutions u e h 2 of p are the critical points of the c functional j u ulvul dx jcf x u dx where f x s jif x t dt in this variational setting the literature usually distinguishes between two situations for prob lem p the subquadratic situation where the potential f satisfies lim suplsl f x s s2 i c m and the superquadratic situation where f satisfies limlsl m f x s s2 00 the main goal of this paper is to present a unified approach to both situations by means of a condition of nonquadraticity at infinity on f more precisely our hypotheses are as follows

space of ricci flows i based on the compactness of the moduli of non collapsed calabi yau spaces with mild singularities we set up a structure theory for polarized kahler ricci flows with proper geo metric bounds our theory is a generalization of the structure theory of non collapsed kahler einstein manifolds as applications we prove the hamilton tian conjecture and the partial c 0 conjecture of tian

local solvability of elliptic and curvature equations on compact manifolds we discuss the search of local constraints for a nonlinear elliptic operator on a compact manifold prove a new version of fredholm theorem for a class of operators with vanishing index and apply it to the local solbability of some prescribed curvature equations

equations du type de monge ampere sur les varietes riemanniennes compactes ii abstract let vn g be a c compact riemannian manifold without boundary given the following changes of metric g g hess l 2 g g 2 hess where a is a fixed constant we study the corresponding monge ampere equations 1 log g g 1 f p 2 log g g 1 f p we first solve eq 2 under some simple assumptions on f c then using an appropriate change of functions that enables us to take advantage of the estimates just carried out for eq 2 we extend to eq 1 all the results proved in our previous articles 5 6 for the usual monge ampere equation although equation 2 is not locally invertible and does not even admit a solution for all f 0 f c v n a similar change of functions leads to partial results about eq 1 via c2 and c3 estimates for eq 2 eventually we give some comments and errata of our previous article p delanoe j funct anal 41 1981 341 353

differential inclusions involving oscillatory terms abstract motivated by mechanical problems where external forces are non smooth we consider the differential inclusion problem d u x f u x g u x in u 0 in u 0 on where r n is a bounded open domain and f and g stand for the generalized gradients of the locally lipschitz functions f and g in this paper we provide a quite complete picture on the number of solutions of d whenever f oscillates near the origin infinity and g is a generic perturbation of order p 0 at the origin infinity respectively our results extend in several aspects those of kristaly and morosanu 2010

radial solutions of dirichlet problems with concave convex nonlinearities abstract we prove the existence of a double infinite sequence of radial solutions for a dirichlet concave convex problem associated with an elliptic equation in a ball of r n we are interested in relaxing the classical positivity condition on the weights by allowing the weights to vanish the idea is to develop a topological method and to use the concept of rotation number the solutions are characterized by their nodal properties

existence of multiple solutions to the equations of monge ampere type abstract in this paper we prove the existence of nontrivial convex solutions to the monge ampere equation det d 2 u x u du with dirichlet or neumann boundary condition where x 0 0 0 we also deal with the existence and multiplicity of convex solutions to the equation det d 2 u x u du our results show that the existence of convex solutions of monge ampere equations is analogous to that of positive solutions of semilinear elliptic equations

a priori estimates for solutions of monge ampere equations on considere les equations de monge ampere elliptiques de la forme ar 2bs ct rt s 2 e pour z z x y definie dans un sous ensemble ouvert du plan x y

a c2 estimate for solutions of complex monge ampere equations on considere le probleme de dirichlet pour des equations de monge ampere elliptiques complexes dans un ensemble ouvert borne pseudoconvexe de c n n 2

local superlinearity and sublinearity for indefinite semilinear elliptic problems in this paper the usual notions of superlinearity and sublinearity for semilinear problems like du f x u are given a local form and extended to indefinite nonlinearities here f x s is allowed to change sign or to vanish for s near zero as well as for s near infinity some of the well known results of ambrosetti bre zis cerami are partially extended to this context

global bifurcation and convex solutions for the monge ampere equation abstract we study the following monge ampere equation det d 2 u 1 n f u in u 0 on by bifurcation technique we establish some results about the existence nonexistence uniqueness and multiplicity of convex solutions for this problem our results generalize and improve many important known results from previous literature

global regularity of solutions of nonlinear second order elliptic and parabolic differential equations on considere des solutions du probleme de dirichlet pour une equation elliptique totalement non lineaire dans un domaine borne lisse et du probleme parabolique analogue

multiple positive solutions for a superlinear problem a topological approach we study the multiplicity of positive solutions for a two point boundary value problem associated to the nonlinear second order equation u f x u 0 we allow x f x s to change its sign in order to cover the case of scalar equations with indefinite weight roughly speaking our main assumptions require that f x s s is below 1 as s 0 and above 1 as s in particular we can deal with the situation in which f x s has a superlinear growth at zero and at infinity we propose a new approach based on topological degree which provides the multiplicity of solutions applications are given for u a x g u 0 where we prove the existence of 2n 1 positive solutions when a x has n positive humps and a x is sufficiently large

multiple positive solutions for the critical kirchhoff type problems involving sign changing weight functions abstract this paper is concerned with the multiplicity of positive solutions for the critical kirchhoff type problems involving indefinite weight functions 0 1 m u 2 d x u q x u q 2 u k x u 2 2 u x u 0 x where is a smooth bounded domain in r n n 3 1 q 2 m s a b s with 0 a 0 and b 0 the weight functions q and k are continuous and changing sign using the nehari manifold fibering maps and ljusternik schnirelmann category we prove that at least two positive solutions for 0 1 exist provided that 1 and 2 4 furthermore by the mountain pass theorem and ekeland s variational principle it is shown that 0 1 possesses at least three positive solutions whenever 2 n 2 including the case that 1 and 2 4 our results generalize some recent results in the literature

a note on a superlinear indefinite neumann problem with multiple positive solutions abstract we prove the existence of three positive solutions for the neumann problem associated to u a t u 1 0 assuming that a t has two positive humps and 0 t a t d t is large enough actually the result holds true for a more general class of superlinear nonlinearities

a multilevel approach to the travelling salesman problem we motivate derive and implement a multilevel approach to the travelling salesman problem the resulting algorithm progressively coarsens the problem initialises a tour and then employs either the lin kernighan lk or the chained lin kernighan clk algorithm to refine the solution on each of the coarsened problems in reverse order in experiments on a well established test suite of 80 problem instances we found multilevel configurations that either improved the tour quality by over 25 as compared to the standard clk algorithm using the same amount of execution time or that achieved approximately the same tour quality over seven times more rapidly moreover the multilevel variants seem to optimise far better the more clustered instances with which the lk and clk algorithms have the most difficulties

le probleme de dirichlet pour des equations de monge ampere complexes modifiees resume let x 2 m g be a strongly pseudoconvex hermitian manifold with c boundary if belongs to c 2 x let us consider the changes of hermitian metric defined by g 1 g i and g 2 g i we solve problems of the form detg i det g 1 f x in x and u on x where f c tx r is an everywhere strictly positive function satisfying some assumptions and u c x

periodic solutions to second order indefinite singular equations abstract the efficient conditions guaranteeing the existence of a t periodic solution to the second order differential equation u h t g u are established in the paper here g is a positive and decreasing function which has a strong singularity at the origin and the weight h l r t z is a sign changing function the obtained results have the form of relation between the multiplicities of the zeroes of the weight function h and the order of the singularity of the nonlinear term the approach is based on leray schauder degree theory proving that no t periodic solution of a certain homotopy appears on the boundary of an unbounded open set during the deformation to an autonomous problem

the complex monge ampere equation on compact kahler manifolds we consider the complex monge ampere equation on a compact kahler manifold m g when the right hand side f has rather weak regularity in particular we prove that estimate of triangle phi and the gradient estimate hold when f is in w 1 p 0 for any p 0 2n as an application we show that there exists a classical solution in w 3 p 0 for the complex monge ampere equation when f is in w 1 p 0

the dirichlet problem for nonlinear second order elliptic equations ii complex monge ampere and uniformaly elliptic equations on considere le probleme de dirichlet pour des equations elliptiques non lineaires d ordre 2 pour une fonction reelle dans un domaine borne de r n a frontiere lisse

three positive solutions for dirichlet problems involving critical sobolev exponent and sign changing weight abstract in this paper we study the decomposition of the nehari manifold via the combination of concave and convex nonlinearities furthermore we use this result and the lusternik schnirelman category to prove that a semilinear elliptic equation involving a sign changing weight function has at least three positive solutions

on some indefinite and non powerlike elliptic equations abstract we consider a class of elliptic problems for the operator v acting on h 0 1 where is a bounded domain the main feature here is the indefinite behavior of both the operator and the nonlinearity based on the global geometry of the energy functional we prove the existence of two local minimizers and a mountain pass critical point for a concave convex type nonlinearity subcritical and critical growth are allowed on the nonlinearity

existence of solutions to a class of semilinear elliptic equations involving general subcritical growth in this paper we consider the semilinear elliptic equation u f x u with the dirichlet boundary value and under suitable assumptions on the nonlinear term f with a more general growth condition some existence results of solutions are given for all 0 via the variational method and some analysis techniques