on the existence of weak solutions of perturbated systems with critical growth sur un domaine borne de r n on considere le probleme de dirichlet pour le systeme elliptique a u g u f sur et u 0 ou a u est un operateur quasi lineaire et la perturbation g u depend de u et de d u on etudie des conditions suffisantes sur a u et sur g u pour l existence de solutions faibles

competition interfaces and second class particles the one dimensional nearest neighbor totally asymmetric simple exclusion process can be constructed in the same space as a last passage percolation model in z 2 we show that the trajectory of a second class particle in the exclusion process can be linearly mapped into the competition interface between two growing clusters in the last passage percolation model using technology built up for geodesics in percolation we show that the competition interface converges almost surely to an asymptotic random direction as a consequence we get a new proof for the strong law of large numbers for the second class particle in the rarefaction fan and describe the distribution of the asymptotic angle of the competition interface

regularizing effect of absorption terms in singular problems abstract we prove existence of solutions to problems whose model is p u u q f u in u 0 in u 0 on where is an open bounded subset of r n n 2 p u is the p laplacian operator for 1 p n q 0 0 and f is a nonnegative function in l m for some m 1 in particular we analyze the regularizing effect produced by the absorption term in order to infer the existence of finite energy solutions in case 1 we also study uniqueness of these solutions as well as examples which show the optimality of the results finally we find local w 1 p solutions in case 1

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

quasilinear elliptic problems with singular and homogeneous lower order terms abstract we deal with singular quasilinear elliptic equations namely u u x u q u q 1 f x in u 0 in u 0 on where is a bounded smooth domain of r n n 3 r 1 q 2 0 l and 0 f l p for some p n 2 we completely describe the set of values of the parameter for which the problem admits solution thus we study existence nonexistence and uniqueness of bounded weak solutions in both cases f 0 and f 0

existence of renormalized solutions to nonlinear elliptic equations with two lower order terms and measure data in this paper we prove the existence of a renormalized solution to a class of nonlinear elliptic problems whose prototype is formula maths where is a bounded open subset of r n n 2 ap is the so called p laplace operator 1 p n is a radon measure with bounded variation on 0 p 1 0 p 1 and c and b belong to the lorentz spaces l p 1 r n p 1 r and l n 1 respectively in particular we prove the existence under the assumptions that p 1 c belongs to the lorentz space ln p 1 r n p 1 r and c n r is small enough

exactly solvable quantum sturm liouville problems the harmonic oscillator with time dependent parameters covers a broad spectrum of physical problems from quantum transport quantum optics and quantum information to cosmology several methods have been developed to quantize this fundamental system such as the path integral method the lewis riesenfeld time invariant method the evolution operator or dynamical symmetry method etc in all these methods solution of the quantum problem is given in terms of the classical one however only few exactly solvable problems of the last one such as the damped oscillator or the caldirola kanai model have been treated the goal of the present paper is to introduce a wide class of exactly solvable quantum models in terms of the sturm liouville problem for classical orthogonal polynomials this allows us to solve exactly the corresponding quantum parametric oscillators with specific damping and frequency dependence which can be considered as quantum sturm liouville problems

on a singular elliptic system with quadratic growth in the gradient abstract in this paper we deal with the following singular elliptic system u u 2 u p p q a x v q u p 2 u f x r n v v 2 v q p q a x u p v q 2 v g x r n where n 3 n 2 4 p q 1 and p q n 2 n 2 we show through the sub and supersolutions method the existence of a nonnegative solution for an approximated system the limit of the approximated solution is a positive solution in the case 0 p q and f g we prove the uniqueness of a solution among others we prove some existence and uniqueness results for some auxiliary problems by using the comparison principle a minimization method and with the help of nehari manifold the proofs rely on the concentration compactness principle

non linear elliptic and parabolic equations involving measure data in this paper we prove the existence of solutions for equations of the type div a du f in a bounded open set u 0 on where a is a possibly non linear function satisfying some coerciveness and monotonicity assumptions and f is a bounded measure we also consider the equation div a du g u f in u 0 on with f l1 or f m u u 0 and the parabolic equivalent of the first elliptic equation

solvability of perturbed elliptic equations with critical growth exponent for the gradient sur un domaine borne cr n on considere le probleme de dirichlet a u g u f u 0 ou a u t 1 n 1 d i a i x u d u definit un operateur pseudomonotone d un espace de sobolev h 0 1 p dans son espace dual et la perturbation g u g x u d u satisfait une condition de croissance

generalized scalar curvature type equations on compact riemannian manifolds the paper is concerned with nonlinear equations of critical sobolev growth involving the p laplace operator these equations generalize the more classical scalar curvature equation

existence results for some quasilinear parabolic equations on considere le probleme de cauchy pour une equation dont la partie principale est un operateur parabolique quasilineaire le terme non lineaire additionnel a au plus une croissance quadratique par rapport au gradient de la solution on demontre l existence d une solution bornee faible sous des hypotheses de regularite minimales en supposant l existence d une sous solution et d une supersolution pour le probleme

holder continuity of the inverse ofp laplacian abstract in this note we study the inverse operator p 1ofp lapalcian on a bounded domain rn we show that p 1 w 1 p w1 p0 is holder continuous and is a compact operator fromv q s tow1 p0 s 0 p q p the conjugate of critical sobolev exponent as an application we study existence of positive solutions of two nonlinear elliptic equations

madelung representation of damped parametric quantum oscillator and exactly solvable schrodinger burgers equations we construct a madelung fluid model with time variable parameters as a dissipative quantum fluid and linearize it in terms of schrodinger equation with time dependent parameters it allows us to find exact solutions of the nonlinear madelung system in terms of solutions of the schrodinger equation and the corresponding classical linear ordinary differential equation with variable frequency and damping for the complex velocity field the madelung system takes the form of a nonlinear complex schrodinger burgers equation for which we obtain exact solutions using complex cole hopf transformation in particular we give exact results for nonlinear madelung systems related with caldirola kanai type dissipative harmonic oscillator collapse of the wave function in dissipative models and possible implications for the quantum cosmology are discussed

lower semicontinuity of functionals via the concentration compactness principle in this paper we show the weak lower semicontinuity of some classes of functionals using the concentration compactness principle of p l lions these functionals involve an integral term and we do not know whether it can be handled by the de giorgi theorem the semicontinuity result allows us to prove several existence results for quasilinear elliptic equations

minimization problem associated with an improved hardy sobolev type inequality abstract we consider the existence or non existence of a minimizer of the following minimization problems associated with an improved hardy sobolev type inequality introduced by ioku 2019 i a inf u w 0 1 p b r 0 b r u p d x b r u p s v a x d x p p s where v a x 1 x s 1 a x r n p p 1 1 x s where 1 p n and 0 s p the minimization problem i a is equivalent to the minimization problem associated with the classical hardy sobolev inequality on r n via a transformation if we restrict ourselves to radial functions in contrast to the classical results for a 0 we show the existence of non radial minimizers for the hardy sobolev critical exponent p s p n s n p on bounded domains furthermore as an application of the transformation we derive an infinite dimensional form of the classical sobolev inequality in some sense

integrable hierarchies and information measures in this paper we investigate integrable models from the perspective of information theory exhibiting various connections we begin by showing that compressible hydrodynamics for a one dimensional isentropic fluid with an appropriately motivated information theoretic extension is described by a general nonlinear schrodinger nls equation depending on the choice of the enthalpy function one obtains the cubic nls or other modified nls equations that have applications in various fields next by considering the integrable hierarchy associated with the nls model we propose higher order information measures which include the fisher measure as their first member the lowest members of the hierarchy are shown to be included in the expansion of a regularized kullback leibler measure while on the other hand a suitable combination of the nls hierarchy leads to a wootters type measure related to a nls equation with a relativistic dispersion relation finally through our approach we are led to construct integrable relativistic nls equations

variations on a strange semi continuity result abstract this paper is devoted to the study of some lower semi continuity results whose model example is lim inf n u n v n u v for any sequence u n v n which weakly converges to u v in h 0 1 2 and satisfies the positivity assumptions u n 0 and v n 0 extensions of this semi continuity result are obtained in the following cases two fixed linear operators two monotone operators two varying linear operators varying domains and non uniformly bounded operators in these various cases the operators under consideration are second order elliptic operators in divergence form and truncation arguments play an essential role finally the case of higher order operators is treated by a quite different approach

exactly solvable hermite laguerre and jacobi type quantum parametric oscillators we introduce exactly solvable quantum parametric oscillators which are generalizations of the quantum problems related with the classical orthogonal polynomials of hermite laguerre and jacobi type introduced in the work of buyukasik et al j math phys 50 072102 2009 quantization of these models with specific damping frequency and external forces is obtained using the wei norman lie algebraic approach this determines the evolution operator exactly in terms of two linearly independent homogeneous solutions and a particular solution of the corresponding classical equation of motion then time evolution of wave functions and coherent states are found explicitly probability densities expectation values and uncertainty relations are evaluated and their properties are investigated under the influence of the external terms

multiplicity of solutions for elliptic quasilinear equations with critical exponent on compact manifolds abstract this paper deals with some perturbation of the so called generalized prescribed scalar curvature type equations on compact riemannian manifolds these equations are nonlinear of critical sobolev growth and involve the p laplacian sufficient conditions are given to have multiple positive solutions

mountain pass solutions for quasi linear equations via a monotonicity trick abstract we obtain the existence of symmetric mountain pass solutions for quasi linear equations without the typical assumptions which guarantee the boundedness of an arbitrary palais smale sequence this is done through a recent version of the monotonicity trick proved in squassina in press 22 the main results are new also for the p laplacian operator

existence and non existence of positive solutions for nonlinear elliptic singular equations with natural growth abstract in this paper we analyze the existence non existence and uniqueness of positive solutions of some nonlinear elliptic equations containing singular terms and natural growth in the gradient term we use an adequate sub super solution method to prove the existence of solutions the characterization of eigenvalues and the integrability of the term u 2 u 2 for the non existence and results from arcoya segura de leon for the uniqueness

on the interface between two growing eden clusters the authors have measured the roughening exponent chi at the interface between a pair of growing eden clusters their numerical results suggest that chi differs from the corresponding exponent for a freely growing interface and that it depends on the initial geometry of the clusters which grow into a vacant sector with an angle theta for theta 180

regularizing effect for some p laplacian systems abstract we study existence and regularity of weak solutions for the following p laplacian system p u a 1 u r 2 u f u w 0 1 p p u r w 0 1 p where is an open bounded subset of r n n 2 p v div v p 2 v is the p laplacian operator for 1 p n a 0 r 1 0 p 1 and f belongs to a suitable lebesgue space in particular we show how the coupling between the equations in the system gives rise to a regularizing effect producing the existence of finite energy solutions

asymptotic behaviour of linear dirichlet parabolic problems with variable operators depending on time in varying domains abstract we study the asymptotic behaviour of the solutions of linear parabolic dirichlet problems when the coefficients and the domains where the problems are posed vary simultaneously in the limit problem it appear the h limit of the operators and as it is usual in the homogenization of dirichlet problems a new term of order zero we also obtain a corrector result

shape optimisation problems governed by nonlinear state equations the purpose of this paper is to give a compactness continuity result for the solution of a nonlinear dirichlet problem in terms of its domain variation the topology in the family of domains is given by the hausdorff metric and continuity is obtained under capacity conditions a generalisation of sverak s result in iv dimensions is deduced as a particular case

strongly nonlinear parabolic equations with natural growth terms in orlicz spaces abstract we prove approximation and compactness results in inhomogeneous orlicz sobolev spaces and look at as an application the cauchy dirichlet equation u a u g x t u u f w 1 x e m where a is a leray lions operator having a growth not necessarily of polynomial type we also give a trace result allowing to deduce the continuity of the solutions with respect to time

ground states of degenerate quasilinear schrodinger equation with vanishing potentials abstract in this paper we study the existence of nontrivial ground state solutions for the following class of p laplacian type equation div a x u v x x p u p 2 u k x x p f u in r n where 1 p n n 3 n p p e 1 d 1 e p p e n p n d p critical hardy sobolev exponent f has a quasicritical growth v and k are nonnegative potentials the function a satisfies a x u c 0 x p h 0 x u p 1 c 0 1 x p h 1 x u p 1 for any r n a e x r n for any two positive functions h 1 l l o c r n h 0 l p p 1 r n with p p

existence and multiplicity of nodal solutions for nonlinear elliptic equations with critical sobolev growth abstract let be a smooth bounded domain of r n n 3 and let a x and f x be two smooth functions defined on a neighbourhood of first we study the existence of nodal solutions for the equation u a x u f x u 4 n 2 u on u 0 on in particular when is a solid torus of r 3 we describe three infinity of pairs of solutions of the equation u u 4 n 2 u on u 0 on afterwards we study the equation when the zero dirichlet condition on the boundary is replaced by a non zero dirichlet condition

the extinction versus the blow up global and non global existence of solutions of source types of degenerate parabolic equations with a singular absorption abstract we consider nonnegative solutions of degenerate parabolic equations with a singular absorption term and a source nonlinear term t u u x p 2 u x x u u 0 f u x t in i 0 t with the homogeneous zero boundary condition on i x 1 x 2 an open bounded interval in r through this paper we assume that p 2 and 0 1 to show the local existence result we prove first a sharp pointwise estimate for u x one of our main goals is to analyze conditions on which local solutions can be extended to the whole time interval t 0 the so called global solutions or by the contrary a finite time blow up 0 0 arises such that lim t 0 u t l i moreover we prove that any global solution must vanish identically after a finite time if provided that either the initial data or the source term is small enough finally we show that the condition f 0 x t 0 x t i 0 is a necessary and sufficient condition for the existence of solution of equations of this type

non equilibrium behaviour of a many particle process density profile and local equilibria one considers a simple exclusion particle jump process on where the underlying one particle motion is a degenerate random walk that moves only to the right one starts with the configuration in which the left halfline is completely occupied and the right one free it is shown that the number of particles at time t between site u t and v t divided by t converges a s to int limits u nu f w dw where f might be called the density profile it is explicitely determined and shown to be an affine function secondly we prove that the distribution of the process looked at by an observer travelling at constant speed u converges weakly to the bernoulli measure with density f u as the time tends to infinity

anisotropic elliptic equations with general growth in the gradient and hardy type potentials abstract in this paper we give existence and regularity results for the solutions of problems whose prototype is q v v h d v q h o x p v p 2 v f x in v 0 on with bounded domain of r n n 2 0 p 1 q p n is a nonnegative continuous function and 0 moreover h is a general norm of r n h o is its polar and q v i 1 n x i h d v p 1 h i d v

multiplicity result for some nonlocal anisotropic equation via nonsmooth critical point theory approach abstract in this paper we study quasilinear problems involving variable exponent growth conditions and nonlocal terms on the whole space r n a multiplicity result is established all the coefficients involved in the terms of the equation depend both on the variable x and the unknown function u our main argument is nonsmooth critical point theory

existence results for some nonlinear elliptic system with lack of compactness abstract we are interested in existence of solutions for some typical elliptic systems governed by two pseudo laplacian operators with critical sobolev exponents we show this result by a local minimization for an adapted variational problem a concentration compactness principle allows us to obtain some convergence results

a quasilinear dirichlet problem with quadratic growth respect to the gradient and l1 data abstract for an open bounded set r n measurable bounded functions a x b x which are strictly positive and p q 0 we prove the existence of a weak solution of the quasilinear b v p div a x u q u x b x u u p 1 u 2 f x x u x 0 x the datum f is assumed to be in l 1 and does not satisfy any sign assumption

existence and nonexistence of solutions for singular quadratic quasilinear equations we study both existence and nonexistence of nonnegative solutions for nonlinear elliptic problems with singular lower order terms that have natural growth with respect to the gradient whose model is 8

courbures scalaires des varietes d invariant conforme negatif in this paper we are interested in the problem of prescribing the scalar curvature on a compact riemannian manifold of negative conformal invariant we give a necessary and sufficient condition when the prescribed function f is nonpositive when sup f 0 we merely find a sufficient condition this is the subject of the first theorem in the second one we prove the multiplicity of the solutions of subcritical for the sobolev imbeddings elliptic equations in another article 8 we will prove the multiplicity of the solutions of the prescribing curvature problem i e for a critical elliptic equation cet article est le premier d une serie de deux on donne ici la demonstration de r6sultats parus dans la note a l academie des sciences 7 on s interesse au probleme de la courbure scalaire prescrite dans le cas d une varite riemannienne d invariant conforme negatif on demontre une condition naecessaire et suffisante pour que le probleme admette une solution lorsque la fonction prescrite f est negative ou nulle lorsque supf 0 on n obtient plus que des conditions suffisantes meme si l on demontre que l une d entre elles est necessaire ces resultats font l objet du theoreme 1 un second theoreme prouve la multiplicite des solutions d equations elliptiques sous critiques relativement aux inclusions de sobolev dans le second article 8 on s interessera a la multiplicite des solutions du probleme de courbure prescrite qui conduit a une equation elliptique dont l exposant est cette fois critique i position du probleme soit m g une variete riemannienne de dimension n n 3 on suppose m compacte et telle que l invariant conforme u soit negatif u inf n 2fiupfm u uo uinf 0 f u2n n 2 n 2 n oii r est la courbure scalaire de m on considere le probleme suivant etant donnee une fonction f continue sur m existe t il une transformation conforme de la metrique g g u4 n 2 g u 0 telle que f soit la courbure scalaire de la variet m g received by the editors march 30 1994 1991 mathematics subject classification primary 53c42 secondary 58g03

nontrivial solutions for perturbations of a hardy sobolev operator on unbounded domains in this paper we study the existence of nontrivial solution of the problem pu d x p u p 2u f u in and u 0 on where rn k is a bounded domain with smooth boundary in rk 1 k n existence is established using mountain pass lemma and concentration of compactness principle

a robin problem for a class of quasilinear operators and a related minimizing problem abstract in this paper we establish the existence of multiple radial solutions for a class of quasilinear operators with nonlinear boundary robin conditions besides other conditions we consider the nonlinearities having a behavior at at least like a linearity of slope less than first eigenvalue 1 r the technical approach is by variational methods which is mainly based on a version of mountain pass theorem due to ghoussoub and preiss

non autonomous perturbations for a class of quasilinear elliptic equations on r abstract this paper is concerned with the existence of two positive solutions for a class of quasilinear elliptic equations on r involving the p laplacian with a non autonomous perturbation the first solution is obtained as a local minimum in a neighborhood of 0 and the second one by a mountain pass argument the special features of the problem here is the complex structure of the linear part which in particular oblige to work into the space w 1 p r then one faces problems in the convergence of the sequences of derivatives u n u

multiple positive solutions for a quasilinear elliptic equation with critical exponential nonlinearity abstract let be a bounded domain in r n n 2 with c 2 boundary in this work we study the existence of multiple positive solutions of the following problem p div u 2 n 2 1 u 2 n u u 2 n 2 1 u 2 n u u p e u n n 1 u 0 in u 2 n 2 1 u 2 n u u q on where 0 p 2 n 1 q 0 n 1 and is a non negative holder continuous function we show that there exists 0 such that p admits at least two solutions in w 1 n for 0 and no solution for

multiplicity results for problems involving the hardy sobolev operator via morse theory we establish some multiplicity results for a class of boundary value problems involving the hardy sobolev operator using morse theory c 2009 elsevier ltd all rights reserved

scale invariant growth processes in expanding space many growth processes lead to intriguing stochastic patterns and complex fractal structures which exhibit local scale invariance properties such structures can often be described effectively by space time trajectories of interacting particles and their large scale behavior depends on the overall growth geometry we establish an exact relation between statistical properties of structures in uniformly expanding and fixed geometries which preserves the local scale invariance and is independent of other properties such as the dimensionality this relation generalizes standard conformal transformations as the natural symmetry of self affine growth processes we illustrate our main result numerically for various structures of coalescing levy flights and fractional brownian motions including also branching and finite particle sizes one of the main benefits of this approach is a full explicit description of the asymptotic statistics in expanding domains which are often nontrivial and random due to amplification of initial fluctuations

hydrodynamical limit for the asymmetric simple exclusion process on demontre une loi forte des grands nombres pour le processus d exclusion simple asymetrique reechelonne

optimal results for parabolic problems arising in some physical models with critical growth in the gradient respect to a hardy potential abstract we deal with the following parabolic problem u t u u p u x 2 f u 0 in 0 t u x t 0 on 0 t u x 0 u 0 x x where r n n 3 is a bounded regular domain such that 0 or r n p 1 0 and f 0 u 0 0 are in a suitable class of functions there are deep differences with respect to the heat equation 0 the main features in the paper are the following if 0 there exists a critical exponent p such that for p p there is no nontrivial local solution p is optimal in the sense that if p p there exists solution for suitable data if we consider the cauchy problem i e r n we find the same phenomenon about the critical power p as above moreover there exists a fujita type exponent f p in the sense that independently of the initial datum for 1 p f any solution blows up in a finite time respect to an integral norm this is a major difference with respect to the heat equation 0

existence and asymptotic behavior of solutions for a class of neumann problems in this paper we study the existence of solutions u w1 p and their asymptotic behavior for the problem e2 pu f u in u 0on where p is the p laplace operator

roughening and inclination of competition interfaces we study the competition interface between two clusters growing over a random vacant sector of the plane in a simple setup which allows us to perform formal computations and obtain analytical solutions we demonstrate that a phase transition occurs for the asymptotic inclination of this interface when the final macroscopic shape goes from curved to noncurved in the first case it is random while in the second one it is deterministic we also show that the flat case stationary growth is a critical point for the fluctuations for curved and flat final profiles the fluctuations are in the kardar parisi zhang kpz scale 2 3 for noncurve final profile the fluctuations are in the same scale of the fluctuations of the initial conditions which in our model are gaussian 1 2

a nonlinear parabolic problem with singular terms and nonregular data abstract we study existence of nonnegative solutions to a nonlinear parabolic boundary value problem with a general singular lower order term and a nonnegative measure as nonhomogeneous datum of the form u t p u h u f in 0 t u 0 on 0 t u u 0 in 0 where is an open bounded subset of r n n 2 u 0 is a nonnegative integrable function p is the p laplace operator is a nonnegative bounded radon measure on 0 t and f is a nonnegative function of l 1 0 t the term h is a positive continuous function possibly blowing up at the origin furthermore we show uniqueness of finite energy solutions in presence of a nonincreasing h

the p dirichlet to neumann operator with applications to elliptic and parabolic problems abstract in this paper we investigate the dirichlet to neumann operator associated with second order quasi linear operators of p laplace type for 1 p which acts on the boundary of a bounded lipschitz domain in r d for d 2 we establish well posedness and holder continuity with uniform estimates of weak solutions of some elliptic boundary value problems involving the dirichlet to neumann operator by employing these regularity results of weak solutions of elliptic problems we show that the semigroup generated by the negative dirichlet to neumann operator on l q enjoys an l q c 0 smoothing effect and the negative dirichlet to neumann operator on the set of continuous functions on the boundary of the domain generates a strongly continuous and order preserving semigroup moreover we establish convergence in large time with decay rates of all trajectories of the semigroup and in the singular case 1 e 2 d d 2 p 2 for some e 0 we give upper estimates of the finite time of extinction

a gradient estimate to a degenerate parabolic equation with a singular absorption term the global quenching phenomena we prove global existence of nonnegative solutions to the one dimensional degenerate parabolic problems containing a singular term we also show the global quenching phenomena for l1 initial datums moreover the free boundary problem is considered in this paper

multiple critical points for nondifferentiable functionals involving hardy potentials abstract in this paper we study general functionals of the calculus of variations with the presence of a hardy potential we will improve several results obtained in the semilinear framework we will first prove a general weak lower semicontinuity result which will imply the existence of a minimum point whenever the functional is coercive then we will demonstrate existence and multiplicity results of critical points even if our functional is not differentiable we will apply a nonsmooth critical point theory developed in corvellec et al nonlinear anal 1 1993 151 and degiovanni and marzocchi ann mat pura appl 167 1994 73

second class particles as microscopic characteristics in totally asymmetric nearest neighbor k exclusion processes summary we study aspects of the hydrodynamics of one dimensional totally asym metric k exclusion building on the hydrodynamic limit of seppalainen 1999 we prove that the weak solution chosen by the particle system is the unique one with maximal current past any fixed location a uniqueness result is needed because we can prove neither differentiability nor strict concavity of the flux function so we cannot use the lax oleinik formula or jump conditions to define entropy solutions next we prove laws of large numbers for a second class particle in k exclusion the macroscopic trajectories of second class particles are characteristics and shocks of the conservation law for the particle density in particular we extend tok exclusion ferrari s result that the second class particle follows a macroscopic shock in the rie mann solution the technical novelty of the proofs is a variational representation for the position of a second class particle in the context of the variational coupling method

universal distributions for growth processes in 1 1 dimensions and random matrices we develop a scaling theory for kardar parisi zhang growth in one dimension by a detailed study of the polynuclear growth model in particular we identify three universal distributions for shape fluctuations and their dependence on the macroscopic shape these distribution functions are computed using the partition function of gaussian random matrices in a cosine potential

some elliptic problems with singular natural growth lower order terms abstract in this paper we deal with a nonlinear elliptic problem whose model is u b d u 2 u f in u 0 on where b 0 and f 0 belongs to a lebesgue space we prove the existence of positive solutions in suitable sobolev spaces depending on f and b

existence of a continuum of solutions for a quasilinear elliptic singular problem abstract in this paper we study the existence of positive solution u h 0 1 for some quasilinear elliptic equations having lower order terms with quadratic growth in the gradient and singularities whose model is u x u 2 u u u p f 0 x x 0 0 p using topological methods we obtain the existence of an unbounded continuum of solutions in the case x is constant we derive the existence of solution for every 0 if 1 2 for any and p 1 even more for l we prove this result if 1 and p 2