emden fowler equations involving critical exponents on considere le probleme y t k y 2 y 2k 3 0 t 0 pour t 2 0 0 soit t inf t 0 y 0 sur t alors t 0 0 on etudie comment la position t du premier zero de y t varie avec

on variational iterative methods for semilinear problems abstract in this article an iterative method suitable for inverting semilinear problems is presented the method employs a two way parametrization that is able to reduce semilinear boundary value problems into simpler and linear systems governed by the laplacian in contrast to newton picard iterations the method prescribes a well defined procedure for constructing initial iterates and thus does not suffer from lack of global convergence with rectangular domains and periodic boundary conditions the technique also allows fast poisson solvers to be engaged theoretical justifications are provided and supported by several experiments numerical results show that the method yields approximations which are comparable to reference solutions

existence and multiplicity of solutions for a nonlocal problem with critical sobolev exponent abstract in this work we are interested in considering the following nonlocal problem a b u 2 d x u u 2 2 u u q 2 u x u 0 x where r n n 4 is a smooth bounded domain a 0 b 0 1 q 2 0 and 2 2 n n 2 is the critical sobolev exponent by using the variational method and the critical point theorem some existence and multiplicity results are obtained

nontrivial solutions of kirchhoff type problems via the yang index abstract we obtain nontrivial solutions of a class of nonlocal quasilinear elliptic boundary value problems using the yang index and critical groups

a stable and conservative finite difference scheme for the cahn hilliard equation we propose a stable and conservative finite difference scheme to solve numerically the cahn hilliard equation which describes a phase separation phenomenon numerical solutions to the equation is hard to obtain because it is a nonlinear and nearly ill posed problem we design a new difference scheme based on a general strategy proposed recently by furihata and mori the new scheme inherits characteristic properties the conservation of mass and the decrease of the total energy from the equation the decrease of the total energy implies boundedness of discretized sobolev norm of the solution this in turn implies by discretized sobolev s lemma boundedness of max norm of the solution and hence the stability of the solution an error estimate for the solution is obtained and the order is o delta x 2 delta t 2 numerical examples demonstrate the effectiveness of the proposed scheme

conservative nonlinear difference scheme for the cahn hilliard equation ii abstract numerical solutions for the cahn hilliard equation is considered using the crank nicolson type finite difference method existence of the solution for the difference scheme has been shown by brouwer fixed point theorem stability convergence and error analysis of the scheme are shown we also show that the scheme preserves the discrete mass even though the linearized scheme in 1 is conditionally stable and does not preserve the mass

the multiplicity of positive solutions for a class of nonlocal elliptic problem abstract in this paper using the theory of fixed point index we prove some results on the multiplicity of positive solutions for a class of nonlocal elliptic problem

local discontinuous galerkin methods with implicit explicit multistep time marching for solving the nonlinear cahn hilliard equation abstract in this paper we develop the fully discrete local discontinuous galerkin ldg methods coupled with the implicit explicit imex multistep time marching for solving the nonlinear cahn hilliard equation to be more specific we rewrite the cahn hilliard equation in a novel form by adding and subtracting a linear term then we discretize the spatial derivatives with the ldg methods and the temporal derivative with the imex multistep method finally a series of numerical experiments are given to verify the accuracy efficiency and validness of proposed method

existence of a positive solution to kirchhoff type problems without compactness conditions abstract the existence of a positive solution to a kirchhoff type problem on r n is proved by using variational methods and the new result does not require usual compactness conditions a cut off functional is utilized to obtain the bounded palais smale sequences

uniqueness and nonuniqueness for positive radial solutions of δu f u r 0 on considere le probleme de l unicite pour les solutions positives du probleme de dirichlet non lineaire u f u x 0 dans u 0 est une boule ou un anneau de r n et f 0 est superlineaire en u et satisfaisait f 0 x 0

on series solution for second order semilinear parabolic ibvps abstract this paper presents a method to obtain semi analytical solutions to second order semilinear ibvps where the standard adomian decomposition may fail to yield nontrivial solutions or fail to produce the correct partial solutions in contrast to the standard adomian decomposition method the proposed solution method is distinguished by simultaneous inversion of the linear differential operators using eigenfunctions expansion representations the proposed method is applied to several examples of initial boundary value problems a linear advection diffusion problem a burgers equation and the deterministic kardar parisi zhang kpz equation it is shown that the dependence of the series solution on the truncation order n the number of eigenfunctions m and the diffusion coefficient is rather complex

existence and multiplicity of solutions for a quasilinear nonlocal problem abstract in this paper we establish two critical points theorems for functional which is not defined on the whole space but c 1 on a closed subset as applications we study the existence and multiplicity of solutions for a class of kirchhoff type equations with bounded nonlocal term

positive solutions for kirchhoff type equations with critical exponent in rn abstract in this paper we study the following nonlinear kirchhoff type equation 0 1 a b r n u 2 d x u u 2 2 u h x x r n u d 1 2 r n where a 0 b 0 n 3 2 2 n n 2 0 and h l 2 2 1 r n 0 is nonnegative by using the variational method we obtain the existence of positive solutions for eq 0 1 under some assumptions on a b

uniqueness theorems for positive radial solutions of quasilinear elliptic equations in a ball we establish a new pohozaev type identity and use it to prove a theorem on the uniqueness of positive radial solutions to the quasilinear elliptic problem div u m 2emsp14 u f u 0 inb andu 0 on b wherebis a finite ball in rn n 3 and 1 0 and 1 p q n 2 n 2 has a unique positive solution whenn 6 this gives a complete answer to an open problem raised by brezis and nirenberg in 1983 in the casen 6 we shall also derive some partial results to the open problem in the casesn 3 4 5

uniqueness of positive solutions of a quasilinear dirichlet problem with exponential nonlinearity uniqueness of positive solutions of the following equation has been obtained where b n is the ball of unit radius and 0 moreover if n 2 then the solutions are nondegenerate

stationary kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity this paper deals with the existence and the asymptotic behavior of non negative solutions for a class of stationary kirchhoff problems driven by a fractional integro differential operator lk and involving a critical nonlinearity in particular we consider the problem m u 2 lku f x u u 2s 2uin u 0in rn where rn is a bounded domain 2s is the critical exponent of the fractional sobolev space hs rn the function f is a subcritical term and is a positive parameter the main feature as well as the main difficulty of the analysis is the fact that the kirchhoff function m could be zero at zero that is the problem is degenerate the adopted techniques are variational and the main theorems extend in several directions previous results recently appeared in the literature

fourth order time stepping for stiff pdes a modification of the exponential time differencing fourth order runge kutta method for solving stiff nonlinear pdes is presented that solves the problem of numerical instability in the scheme as proposed by cox and matthews and generalizes the method to nondiagonal operators a comparison is made of the performance of this modified exponential time differencing etd scheme against the competing methods of implicit explicit differencing integrating factors time splitting and fornberg and driscoll s sliders for the kdv kuramoto sivashinsky burgers and allen cahn equations in one space dimension implementation of the method is illustrated by short matlab programs for two of the equations it is found that for these applications with fixed time steps the modified etd scheme is the best

a further consideration of partial solutions in the decomposition method abstract reference 1 showed that in partial differential equation problems involving linear operator terms with respect to x y z t considerable work is saved by solving only one of the linear operator equations to obtain a so called partial solution the heat equation was shown as a simple example which appeared to be a counter example because one partial solution was identically zero the l x u operator equation led to a zero solution while the l t u operator equation led to the correct solution e t sin x this is a case of asymptotic equality as t rather than equality it arises when boundary conditions are special as discussed in 1 in cases where one operator annihilates the series in a finite number of terms one partial solution may not satisfy the given conditions this paper provides further insight into cases where such an improper solution can be obtained and the resolution of such cases by appropriate expansion of the initial term without making a priori assumptions about the solution

the fourier pseudospectral method with a restrain operator for the korteweg de vries equation abstract in this paper we develop a fourier pseudospectral method with a restrain operator for the korteweg de vries equation we prove the generalized stability of the schemes and give convergence estimations depending on the smoothness of the solution of the p d e

morse index and uniqueness for positive solutions of radial p laplace equations we study the positive radial solutions of the dirichlet problem p u f u 0 in b u 0 in b u 0 on b where p u div u p 2 u p 1 is the p laplace operator b is the unit ball in r n centered at the origin and f is a c 1 function we are able to get results on the spectrum of the linearized operator in a suitable weighted space of radial functions and derive from this information on the morse index in particular we show that positive radial solutions of mountain pass type have morse index 1 in the subspace of radial functions of w 1 p 0 b we use this to prove uniqueness and nondegeneracy of positive radial solutions when f is of the type u s u q and p 2

uniqueness theorems for negative radial solutions of k hessian equations in a ball abstract in this paper we apply a pohozaev type identity and the monotone separation techniques to prove the uniqueness of negative radial solutions of k hessian equations in a finite ball

revisit on partial solutions in the adomian decomposition method solving heat and wave equations the adomian decomposition method is considered in application to heat and wave equations the so called partial solution technique is used it is shown that the fundamental equation of the method is well defined only for certain types of boundary conditions in cases involving inhomogeneous boundary conditions improper results may be obtained by former method this paper presents a further insight into partial solutions in the decomposition method and the resolution of such cases

the fourier pseudospectral method with a restrain operator for the rlw equation in this paper we develop a new fourier pseudospectral method with a restrain operator for the modified korteweg de vries mkdv equation we prove the generalized stability of the scheme from which the convergence follows

on the existence and multiplicity of solutions for a class of singular elliptic problems abstract in this paper we prove the existence and multiplicity of solutions for a class of elliptic problems with singular coefficients we also prove the existence of a sign changing solution for a brezis nirenberg type problem with hardy potential and singular coefficients our proofs are based on the krasnoselskii genus the concentration compactness principle of lions and the linking theorem

positive solutions for a quasilinear elliptic equation of kirchhoff type this paper is concerned with the existence of positive solutions to the class of nonlocal boundary value problems of the type m w u 2dx du f x u email protected u 0 email protected email protected where w is a smooth bounded domain of n m is a positive function and f has subcritical growth

concentration phenomena for a class of fractional kirchhoff equations in rn with general nonlinearities abstract in this paper we study the following class of fractional kirchhoff problems e 2 s m e 2 s n u s 2 s u v x u f u in r n u h s r n u 0 in r n where e 0 is a small parameter s 0 1 n 2 s is the fractional laplacian v r n r is a positive continuous function m 0 r is a kirchhoff function satisfying suitable conditions and f r r fulfills berestycki lions type assumptions of subcritical or critical type using suitable variational arguments we prove the existence of a family of positive solutions u e which concentrates at a local minimum of v as e 0

multiple positive solutions for kirchhoff type of problems with singularity and critical exponents abstract in this paper we study multiplicity of positive solutions for a class of kirchhoff type of equations with the nonlinearity containing both singularity and critical exponents we obtain two positive solutions via the variational and perturbation methods

spectral methods for the euler equations i fourier methods and shock capturing spectral methods for compressible flows are introduced in relation to finite difference and finite element techniques within the framework of the method of weighted residuals current spectral collociation methods are put into historical context the basic concepts of fourier spectral collocation methods are provided filtering strategies for shock capturing approaches are also presented fourier shock capturing techniques are evaluated using a one dimensional periodic astrophysical nozzle problem

infinitely many radial solutions for kirchhoff type problems in rn abstract in this paper we concern with a class of kirchhoff type problems in r n by using the fountain theorem we obtain three existence results of infinitely many radial solutions for the problem

global bifurcation and positive solution for a class of fully nonlinear problems in this paper we study global bifurcation phenomena for the following kirchhoff type problem m u x 2 d x u f x u in u 0 on where m is a continuous function under some natural hypotheses we show that 1 a m 0 0 is a bifurcation point and there is a global continuum c emanating from 1 a m 0 0 where 1 a denotes the first eigenvalue of the above problem with f x s a x s as an application of the above result we study the existence of positive solution for this problem with asymptotically linear nonlinearity

infinitely many positive solutions for kirchhoff type problems abstract this paper is concerned with the existence of infinitely many positive solutions to a class of kirchhoff type problem a b u 2 d x u f x u in and u 0 on where is a smooth bounded domain of r n a b 0 0 and f r r is a caratheodory function satisfying some further conditions we obtain a sequence of a e positive weak solutions to the above problem tending to zero in l with f being more general than that of k perera z zhang nontrivial solutions of kirchhoff type problems via the yang index j differential equations 221 2006 246 255 z zhang k perera sign changing solutions of kirchhoff type problems via invariant sets of descent flow j math anal appl 317 2006 456 463

the nehari manifold for a kirchhoff type problem involving sign changing weight functions abstract this paper examines a class of kirchhoff type equations that involve sign changing weight functions using nehari manifold and fibering map the existence of multiple positive solutions is established

study of a nonlinear kirchhoff equation with non homogeneous material abstract in this paper we study a non homogeneous elliptic kirchhoff equation with nonlinear reaction term we analyze the existence and uniqueness of positive solution the main novelty is the inclusion of non homogeneous term making the problem without a variational structure we use mainly bifurcation arguments to get the results

remarks on an elliptic equation of kirchhoff type abstract we present a short survey on the nonlocal elliptic boundary value problem m u 2 d x u f x u in u 0 on where is a smooth bounded domain of r n m is a positive function and f has subcritical growth

stability analysis of large time stepping methods for epitaxial growth models numerical methods for solving the continuum model of the dynamics of the molecular beam epitaxy mbe require very large time simulation and therefore large time steps become necessary the main purpose of this work is to construct and analyze highly stable time discretizations which allow much larger time steps than those of a standard implicit explicit approach to this end an extra term which is consistent with the order of the time discretization is added to stabilize the numerical schemes then the stability properties of the resulting schemes are established rigorously numerical experiments are carried out to support the theoretical claims the proposed methods are also applied to simulate the mbe models with large solution times the power laws for the coarsening process are obtained and are compared with previously published results

uniqueness and profile of positive solutions of a critical exponent problem with hardy potential abstract we obtain the uniqueness and blow up profile of positive least energy solution of a critical exponent problem with hardy potential in a ball

an elementary proof of the uniqueness of positive radial solutions of a quasilinear dirichlet problem we consider the quasilinear elliptic equation text div left nabla u m 2 nabla u right u p lambda u m 2 u in in b where b is a ball or an annulus in n 1 m n p is a positive real number and e using a generalized poho aev type variational identity of ni serrin or pucci and serrin and an elementary calculus lemma we establish uniqueness of positive radial solutions for the dirichlet boundary condition if either left text i right b text is a ball lambda underset raise0 3em hbox smash scriptscriptstyle geqslant text 0 1 p underset raise0 3em hbox smash scriptscriptstyle leqslant frac mn n m text for m n text and 1 p infty text for m n text or left text ii right b text is an annulus lambda in mathbb r text and p frac mn n m

a direct variational methods applied to burgers equation in this study a compact approximate method in limiting form for calculating the solution of burgers equation with appropriate boundary conditions is presented the results obtained by present method are found to be in good agreement with those due to earlier authors and offers appreciable advantages for burgers like nonlinear problems

a uniqueness result for a nonlocal equation of kirchhoff type and some related open problem abstract we study the uniqueness of solution for the following boundary value problem involving a nonlocal equation of kirchhoff type a b u 2 d x u f u in u 0 here is a bounded open set in r n with smooth boundary a b are positive real numbers and f r r is a continuous function in particular we give an answer to an open problem recently proposed by b ricceri

on a p kirchhoff equation via fountain theorem and dual fountain theorem abstract in this paper we show the existence of infinite solutions to the kirchhoff type quasilinear elliptic equation m u p x u p d x p 1 p u x u p 2 u f x u in a smooth bounded domain r n with nonlinear boundary condition u p 2 u u p 2 on the method we used here is based on fountain theorem and dual fountain theorem

a uniqueness result for a semilinear elliptic problem a computer assisted proof abstract starting with the famous article a gidas w m ni l nirenberg symmetry and related properties via the maximum principle comm math phys 68 1979 209 243 many papers have been devoted to the uniqueness question for positive solutions of u u u p in u 0 on where p 1 and ranges between 0 and the first dirichlet eigenvalue 1 of for the case when is a ball uniqueness could be proved mainly by ode techniques but very little is known when is not a ball and then only for 0 in this article we prove uniqueness for all 0 1 in the case 0 1 2 and p 2 this constitutes the first positive answer to the uniqueness question in a domain different from a ball our proof makes heavy use of computer assistance we compute a branch of approximate solutions and prove existence of a true solution branch close to it using fixed point techniques by eigenvalue enclosure methods and an additional analytical argument for close to 1 we deduce the non degeneracy of all solutions along this branch whence uniqueness follows from the known bifurcation structure of the problem

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

the restrictions and improvement of the adomian decomposition method the adomian decomposition method is a creative and effective method for exactly solving functional equations of various kinds in this paper the principle of the decomposition method is described and its restrictions are discussed then an improvement is proposed which overcomes the restrictions and complements the ability of the adomian decomposition method some concrete examples are also studied to illustrate the validity of the improvement

error estimates with smooth and nonsmooth data for a finite element method for the cahn hilliard equation a finite element method for the cahn hilliard equation a semilinear parabolic equation of fourth order is analyzed both in a spatially semidiscrete case and in a completely discrete case based on the backward euler method error bounds of optimal order over a finite time interval are obtained for solutions with smooth and nonsmooth initial data a detailed study of the regularity of the exact solution is included the analysis is based on local lipschitz conditions for the nonlinearity with respect to sobolev norms and the existence of a ljapunov functional for the exact and the discretized equations is essential a result concerning the convergence of the attractor of the corresponding approximate nonlinear semigroup upper semicontinuity with respect to the discretization parameters is obtained as a simple application of the nonsmooth data error estimate

existence of a positive solution for a kirchhoff problem type with critical growth via truncation argument abstract in this paper we shall show an existence result of a positive solution for a kirchhoff problem type in a bounded domain of r n that is for the problem m u 2 d x u f x u u 2 2 u in u 0 on we shall study the asymptotic behavior of this solution when converges to infinity our approach is based on the variational method an appropriated truncated argument and a priori estimates to obtain the solution

implicit explicit methods for time dependent partial differential equations implicit explicit imex schemes have been widely used especially in conjunction with spectral methods for the time integration of spatially discretized partial differential equations pdes of diffusion convection type typically an implicit scheme is used for the diffusion term and an explicit scheme is used for the convection term reaction diffusion problems can also be approximated in this manner in this work we systematically analyze the performance of such schemes propose improved new schemes and pay particular attention to their relative performance in the context of fast multigrid algorithms and of aliasing reduction for spectral methods for the prototype linear advection diffusion equation a stability analysis for first second third and fourth order multistep imex schemes is performed stable schemes permitting large time steps for a wide variety of problems and yielding appropriate decay of high frequency error modes are identified numerical experiments demonstrate that weak decay

a pseudospectral scheme for the numerical calculation of shocks abstract a pseudospectral numerical scheme is developed in order to calculate the propagation of a shock wave we use a two step time differencing method and a chebyshev transform method to compute the space derivative such a scheme has been used to reduce the oscillations due to gibb s phenomenon this numerical method is applied to the solution of the burgers equation without viscosity term the accuracy of the numerical solutions is compared to the one given by two finite difference methods

a nonconforming finite element method for the two dimensional cahn hilliard equation the cahn hilliard equation is a nonlinear evolutionary equation that is fourth order in space in this paper a continuous in time finite element galerkin approximation is considered we use the nonconforming morley element and derive optimal order error bounds in l 2

semilinear elliptic equations and supercritical growth where p p the aim of the present paper is to understand for the case p pc the behaviour of solutions to 1 i with large supremum norm to this end 0 is restricted to be the unit ball f3c r3 as in this case the behaviour of solutions to 1 1 can be described in some detail according to the results proved in gidas ni and nirenberg s all positive solutions of 1 1 must be radial when q b with the scaling iid ilip in the case n 3 problem 1 1 reduces to the following ordinary differential equation problem

on ground state solutions for a kirchhoff type equation with critical growth in this paper with the aid of variational method and concentration compactness principle a positive ground state solution is obtained for a class of kirchhoff type equations with critical growth a b r 3 u 2 d x u u 5 k x u q 1 x r 3 u d 1 2 r 3 where 0 is a parameter 2 q 6 and k satisfies some conditions

positive solutions to some equations with homogeneous operator abstract in this paper we discuss the positive solutions to the equation u u a a u b u u 0 where a is a positive linear completely continuous operator b is an homogeneous operator defined on a cone in a real banach space and u a b u by using the fixed point index theory when u 0 is sufficiently small the spectral radius r a 1 and 1 where sgn b we obtain a positive solution to the above equation under some appropriate conditions the new results generalize the previous research about the homogeneous operator equation as an application by using our main theorem we can obtain a symmetrical positive solution to the one dimensional kirchhoff equation

multiplicity of positive solutions for p laplace equation with superlinear type nonlinearity abstract in this work we consider the problem p r p u u p u q u 0 in b r 0 u 0 on b r 0 where p 2 p np n p and b r 0 is the open ball of radius r centered about the origin in r n n 3 under the condition that p 1 q p 2 n p 1 we show that there exists r 0 such that p r admits atleast two solutions for all r r and no solution if r this extends the results of nonlinear anal tma 10 8 1986 755 to the case of p laplace equation

infinitely many positive solutions for a p x kirchhoff type equation this paper is concerned with the existence of infinitely many positive solutions to a class of p x kirchhoff type problem by means of a direct variational approach and the theory of the variable exponent sobolev spaces we establish the existence of infinitely many distinct positive solutions whose w1 p x norms and l norms tend to zero under suitable hypotheses about nonlinearity

berestycki lions conditions on ground state solutions for kirchhoff type problems with variable potentials based on the strategy introduced by chen and tang adv nonlinear anal 9 496 515 2020 and some new tricks we prove that the nonlinear problem of kirchhoff type a b r3 u 2dx u v x u f u x r3 in h1 r3 admits two classes of ground state solutions under the general berestycki lions assumptions on the nonlinearity f which are almost necessary conditions as well as some weak assumptions on the potential v moreover we also give a simple minimax characterization of the ground state energy our results improve and extend recent theorems in several directions

existence results of positive solutions of kirchhoff type problems abstract in the present paper we use variational methods to prove two existence results of positive solutions of the following kirchhoff type problems a b u 2 u f x u in u 0 on one deals with the asymptotic behaviors of f near zero and infinity and the other deals with 4 superlinear of f at infinity

uniqueness of positive solutions of a n laplace equation in a ball in rn with exponential nonlinearity abstract let n 2 and r n be a bounded domain then by trudinger moser embedding w 0 1 n is embedded in an orlicz space consisting of exponential functions consider the corresponding semilinear n laplace equation with critical or sub critical exponential nonlinearity in a ball b r with dirichlet boundary condition in this paper we prove that under suitable growth conditions on the nonlinearity there exists an 0 0 and a corresponding r 0 0 0 such that for all 0 r r 0 the problem admits a unique nondegenerate positive radial solution u with u 0

global nonexistence for nonlinear kirchhoff systems in this paper we consider the problem of non continuation of solutions of dissipative nonlinear kirchhoff systems involving the p x laplacian operator and governed by nonlinear driving forces f f t x u as well as nonlinear external damping terms q q t x u u t both of which could significantly dependent on the time t the theorems are obtained through the study of the natural energy eu associated to the solutions u of the systems thanks to a new approach of the classical potential well and concavity methods we show the nonexistence of global solutions when the initial energy is controlled above by a critical value that is when the initial data belong to a specific region in the phase plane several consequences interesting in applications are given in particular subcases the results are original also for the scalar standard wave equation when p 2 and even for problems linearly damped

a fourier series approach to burgers equation burgers equation is known to have wide application and attempts have been made in the past to solve the non homogeneous form of the equation because of the limitations of the analytical solution for certain values of the parameter nu due to slow convergence the authors discuss a numerical approach which uses the method of lines to solve the homogeneous case and involves finite fourier series the method is attractive as it works efficiently and provides good results for nu 1 0 and results with non increasing energy for nu 0 1

equality of partial solutions in the decomposition method for linear or nonlinear partial differential equations abstract we consider the solution of partial differential equations for initial boundary conditions using the decomposition method the partial solutions obtained from the seperate equations for the highest ordered linear operator terms are shown to be identical when the boundary conditions are general and asymptotically equal when the boundary conditions in one independent variable are independet of other variables

the sign changing solutions for nonlinear elliptic problem with carrier type abstract in this paper using the fixed point index method we are concerned with the existence of sign changing solutions of the following problem 0 1 a b u x d x u f u x in u x 0 x on where a 0 b 0 and r n is a bounded open set with smooth boundary and 1

positive least energy solutions and phase separation for coupled schrodinger equations with critical exponent in this paper we study the following coupled schrodinger system which can be seen as a critically coupled perturbed brezis nirenberg problem left begin array ll delta u lambda 1 u mu 1 u 3 beta uv 2 quad x in omega delta v lambda 2 v mu 2 v 3 beta vu 2 quad x in omega u geqq 0 v geqq 0 rm in omega quad u v 0 quad rm on partial omega end array right here omega subset mathbb r 4 is a smooth bounded domain lambda 1 omega 0 and beta neq 0 where lambda 1 omega is the first eigenvalue of with the dirichlet boundary condition note that the nonlinearity and the coupling terms are both critical in dimension 4 that is frac 2n n 2 4 when n 4 we show that this critical system has a positive least energy solution for negative positive small and positive large for the case in which lambda 1 lambda 2 we obtain the uniqueness of positive least energy solutions we also study the limit behavior of the least energy solutions in the repulsive case beta to infty and phase separation is expected these seem to be the first results for this schrodinger system in the critical case

multispike solutions for the brezis nirenberg problem in dimension three abstract we consider the problem u u u 5 0 u 0 in a smooth bounded domain in r 3 under zero dirichlet boundary conditions we obtain solutions to this problem exhibiting multiple bubbling behavior at k different points of the domain as tends to a special positive value 0 which we characterize in terms of the green function of

a sub supersolution approach for a quasilinear kirchhoff equation in this paper we establish an existence result for a quasilinear kirchhoff equation via a sub and supersolution approach by using the minty browder s theorem for pseudomonotone operators theory

a note on a nonlinear elliptic problem with a nonlocal coefficient abstract in this paper we investigate a nonlocal and nonlinear elliptic problem p a u 2 d x u u u p in u 0 on where n 3 r n is a bounded domain with smooth boundary a is a nondegenerate continuous function p 1 and r we show several effects of the nonlocal coefficient a on the structure of the solution set of p we first introduce a scaling observation and describe the solution set by using that of an associated semilinear problem this allows us to get unbounded continua of solutions u of p a rich variety of new bifurcation and multiplicity results are observed we also prove that the nonlocal coefficient can induce up to uncountably many solutions in a convenient way lastly we give some remarks from the variational point of view

a critical kirchhoff type problem involving a nonlocal operator abstract in this paper we show the existence of non negative solutions for a kirchhoff type problem driven by a nonlocal integrodifferential operator that is m u z 2 l k u f x u u 2 2 u in u 0 in r n where l k is an integrodifferential operator with kernel k is a bounded subset of r n m and f are continuous functions z is a functional norm and 2 is a fractional sobolev exponent

energy ssp imex runge kutta methods for the cahn hilliard equation we consider the cahn hilliard equation which describes phase separation phenomenon the equation is discretized by using a fourth order compact difference scheme in space and strong stability preserving ssp implicit explicit imex runge kutta methods in time the new methods have two distinct features 1 the large time steps can be used in the numerical simulation because the energy is stable and 2 the energy functional decreases by time unconditional energy stability of first order and second order methods are proved numerical experiments are given to demonstrate the performance of the proposed methods

the critical neumann problem of kirchhoff type in this paper we study the critical neumann problem of kirchhoff type by using the variational method and the concentration compactness argument the existence and multiplicity of nontrivial solutions are obtained

two positive solutions for the kirchhoff type elliptic problem with critical nonlinearity in high dimension abstract we investigate the kirchhoff type elliptic problem with critical nonlinearity 1 u 2 d x u u q u 2 1 u 0 in u 0 on where n 5 r n is a bounded domain with smooth boundary 0 r 2 2 n n 2 and q 1 2 1 we prove the existence of two solutions of it via the variational method since n 5 and 0 the uniqueness assertion for the associated limiting problem may fail this causes serious difficulties in controlling concentrating palais smale sequences we overcome these by introducing new techniques for a mountain pass type solution we utilize the limit function of the fibering maps of the concentrating palais smale sequence this tool is based on our careful setting of nehari type sets on the other hand a suitable modification to a concentrating minimizing sequence enables us to obtain a global minimum solution this is the first work which proves the multiplicity of positive solutions of the kirchhoff type critical problem in high dimension

sign changing solutions of kirchhoff type problems via invariant sets of descent flow we obtain sign changing solutions of a class of nonlocal quasilinear elliptic boundary value problems using variational methods and invariant sets of descent flow

sign changing and multiple solutions of kirchhoff type problems without the p s condition abstract the existence of nontrivial solutions of kirchhoff type equations is an important nonlocal quasilinear problem in this paper we use minimax methods and invariant sets of descent flow to prove two interesting existence theorems for the following 4 superlinear kirchhoff type problems without the p s condition one concerning the existence of a nontrivial solution and the other one concerning the existence of sign changing solutions and multiple solutions a b u 2 u f x u in u 0 on

existence and multiplicity of solutions for kirchhoff type equations abstract in this paper the existence and multiplicity of weak solutions are obtained for a class of kirchhoff type problems with dirichlet boundary value conditions by using the mountain pass theorem the local linking theorem the fountain theorem and the symmetric mountain pass lemma in critical point theory

second order implicit explicit total variation diminishing schemes for the euler system in the low mach regime abstract in this work we consider the development of implicit explicit total variation diminishing tvd methods also termed ssp strong stability preserving for the compressible isentropic euler system in the low mach number regime the proposed scheme is asymptotically stable with a cfl condition independent of the mach number in addition it degenerates in the low mach number regime to a consistent discretization of the incompressible system since it has been proved that implicit schemes of order higher than one cannot be tvd ssp 30 we construct a new paradigm of implicit time integrators by coupling first order in time schemes with second order ones in the same spirit as highly accurate shock capturing tvd methods in space for this particular class of schemes the tvd property is first proved on a linear model advection equation and then extended to the isentropic euler case the result is a method which interpolates from the first to the second order both in space and time it preserves the monotonicity of the solution and is highly accurate for all choices of the mach number moreover the time step is only restricted by the non stiff part of the system we finally show thanks to one and two dimensional test cases that the method indeed possesses the claimed properties

the critical problem of kirchhoff type elliptic equations in dimension four abstract we study the following kirchhoff type elliptic problem p a b u 2 d x u u q u 3 u 0 in u 0 on where r 4 is a bounded domain with smooth boundary moreover we assume a 0 b 0 and 1 q 3 in this paper we prove the existence of solutions of p our tools are the variational method and the concentration compactness argument for ps sequences

spectral methods for hyperbolic problems we review the current state of fourier and chebyshev collocation methods for the solution of hyperbolic problems with an eye to basic questions of accuracy and stability of the numerical approximations throughout the discussion we emphasize recent developments in the area such as spectral penalty methods the use of filters the resolution of the gibbs phenomenon and issues related to the solution of nonlinear conservations laws such as conservation and convergence we also include a brief discussion on the formulation of multi domain methods for hyperbolic problems and conclude with a few examples of the application of pseudospectral collocation methods for solving nontrivial systems of conservation laws

local discontinuous galerkin methods for the cahn hilliard type equations in this paper we develop local discontinuous galerkin ldg methods for the fourth order nonlinear cahn hilliard equation and system the energy stability of the ldg methods is proved for the general nonlinear case numerical examples for the cahn hilliard equation and the cahn hilliard system in one and two dimensions are presented and the numerical results illustrate the accuracy and capability of the methods