existence of sign changing solutions for a nonlinear elliptic problem on rn with a hardy term abstract let n 3 1 p n 2 n 2 and 0 n 2 2 4 we study the existence of multiple sign changing solutions of u h 1 r n u u u x 2 u p 1 u in r n which are not radially symmetric

some further results for a class of weighted nonlinear elliptic equations following our previous work m lin multiple positive solutions for a class of weighted nonlinear elliptic equations preprint 2006 we give an exact growth order near zero for positive solutions of a class of weighted elliptic equations and use it to give the existence of multiple positive solutions and sign changing solutions of the considered problem

positive solutions for some singular critical growth nonlinear elliptic equations let omega be a bounded domain in r n n greater than or equal to 4 with smooth boundary partial derivativeomega and the origin 0 is an element of omega mu n 2 2 2 2 2n n 2 k x is a smooth function on omega and positive somewhere we obtain existence results of positive solutions to the dirichlet problem deltau mu u x 2 k x u 2 2 u f x u on omega u 0 on partial derivativeomega for various k x and suitable number mu c 2004 elsevier ltd all rights reserved

pseudo parabolic regularization of forward backward parabolic equations power type nonlinearities we study a quasilinear parabolic equation of forward backward type under assumptions on the nonlinearity which hold for a wide class of mathematical models using a pseudo parabolic regularization of power type we prove existence and uniqueness of positive solutions of the regularized problem in a space of radon measures it is shown that these solutions satisfy suitable entropy inequalities we also study their qualitative properties in particular proving that the singular part of the solution with respect to the lebesgue measure is constant in time

a pragmatic approach to the problem of the self adjoint extension of hamilton operators with the aharonov bohm potential we consider the problem of the self adjoint extension of hamilton operators for charged quantum particles in the pure aharonov bohm potential infinitely thin solenoid we present a pragmatic approach to the problem based on the orthogonalization of the radial solutions for different quantum numbers then we discuss a model of a scalar particle with a magnetic moment which allows us to explain why the self adjoint extension contains arbitrary parameters and give a physical interpretation

existence of a nontrivial weak solution to quasilinear elliptic equations with singular weights and multiple critical exponents abstract in this paper we consider the existence of a non trivial weak solution to a quasilinear elliptic equation with singular weights and multiple critical exponents in the whole space firstly we get the existence of a local palais smale sequence by verifying the geometric conditions of the mountain pass lemma secondly we study the concentration properties of the palais smale sequence of a zero weak limit thirdly we deduce by contradiction to eliminate the possibility of a zero weak limit case lastly applying a monotonic inequality we shall prove that the nontrivial weak limit of the palais smale sequence is indeed a weak solution

interactions of solitons in nonintegrable systems direct perturbation method and applications abstract problems of approximate description of soliton interaction in nonintegrable systems are discussed it is shown that in an arbitrary lagrangian system in the first nontrivial order of the small parameter asymptotic scheme for a system of solitons with close velocities a lagrangian may be introduced analogous to that for classical particles with a pair interaction potential a classification of possible types of soliton interaction is given and criteria of the existence of bound states are formulated a possibility of generalization of approximated description of soliton interaction in the multidimensional case is noted

on the second solution to a critical growth robin problem abstract we investigate the existence of the second mountain pass solution to a robin problem where the equation is at critical growth and depends on a positive parameter more precisely we determine existence and nonexistence regions for this type of solutions depending both on and on the parameter in the boundary conditions

spectral methods in time for a class of parabolic partial differential equations in this paper we introduce a fully spectral solution for the partial differential equation u t uu x vu x x mu x x x lu x x x x 0 for periodic boundary conditions in space the use of fourier expansion in x admits of a particularly efficient algorithm with respect to expansion of the time dependence in a chebyshev series boundary conditions other than periodic may still be treated with reasonable though lesser efficiency for all cases very high accuracy is attainable at moderate computational cost relative to the expense of variable order finite difference methods in time

positive solutions for elliptic equations involving critical sobolev exponents and hardy terms with neumann boundary conditions abstract in this paper neumann problem for nonlinear elliptic equations with critical sobolev exponents and hardy terms is studied by variational method based on the variant of the mountain pass theorem of ambrosetti and rabinowitz without ps condition we prove the existence of positive solutions

some further results on a semilinear equation with concave convex nonlinearity abstract following our previous work j math anal appl 295 2004 341 we give an exact growth order near zero for positive solutions of a class of elliptic equation and use it to give the existence of multiple solutions with negative energy multiple positive solutions and sign changing solutions of the considered problem even in the particular case our results extend a previous work by ambrosetti garcia peral j funct anal 137 1996 219

interaction between the geometry of the boundary and positive solutions of a semilinear neumann problem with critical nonlinearity abstract we consider the problem u u un 2 n 2 u 0 in u v 0 on where is a bounded smooth domain in r n n 3 we show that for large least energy solutions of the above problem have a unique maximum point p on and the limit points of p as are contained in the set of the points of maximum mean curvature we also prove that if has k peaks then the equation has at least k solutions for large

sign changing solutions for elliptic problems with critical sobolev hardy exponents abstract let r n be a smooth bounded domain such that 0 n 7 0 s 2 s 2 n s n 2 we prove the existence of sign changing solutions for the singular critical problem u u x 2 u 2 s 2 x s u u with dirichlet boundary condition on for suitable positive parameters and

resolvent convergence in norm for dirac operator with aharonov bohm field we consider the hamiltonian for relativistic particles moving in the aharonov bohm magnetic field in two dimensions the field has like singularity at the origin and the hamiltonian is not necessarily essentially self adjoint the self adjoint realization requires one parameter family of boundary conditions at the origin we approximate the point like field by smooth ones and study the problem of norm resolvent convergence to see which boundary condition is physically reasonable among admissible boundary conditions we also study the effect of perturbations by scalar potentials roughly speaking the obtained result is that the limit self adjoint realization is different even for small perturbation of scalar potentials according to the values of magnetic fluxes it changes at half integer fluxes the method is based on the resolvent analysis at low energy on magnetic schrodinger operators with resonance at zero energy and the resonance plays an important role from a mathematical point of view however it has been neglected in earlier physical works the emphasis here is placed on this natural aspect

on the elliptic problems with critical weighted sobolev hardy exponents abstract let r n be a smooth bounded domain such that 0 n 3 in this paper we deal with the conditions that ensure the existence of nontrivial solutions for the elliptic equation div x 2 a u u x 2 1 a u p 2 x b p u u with dirichlet boundary condition which involving the caffarelli kohn nirenberg inequalities the results depend crucially on the parameters a b and

aharonov bohm effect with δ type interaction a quantum particle interacting with a thin solenoid and a magnetic flux is described by a five parameter family of hamilton operators obtained via the method of self adjoint extensions one of the parameters the value of the flux corresponds to the aharonov bohm effect the other four parameters correspond to the strength of a singular potential barrier the spectrum and eigenstates are computed and the scattering problem is solved

estimates of the conformal scalar curvature equation via the method of moving planes in this paper we derive a local estimate of a positive singular solution u near its singular set z of the conformal equation where k x is a positive continuous function z is a compact subset of and g satisfies that is nonincreasing for t 0 assuming that the order of flatness at critical points of k on z is no less than we prove that through the application of the method of moving planes the inequality holds for any solution of 0 1 with cap z 0 by the same method we also derive a harnack type inequality for smooth positive solutions let u satisfy assume that the order of flatness at critical points of k is no less than n 2 then the inequality holds for r 1 we also show by examples that the assumption about the flatness at critical points is optimal for validity of the inequality 0 4 1997 john wiley sons inc

chaotic behaviour of soliton lattice in an unstable dissipative dispersive nonlinear system abstract dynamical behaviours of a sequence of asymmetric soliton like pulses arising in an unstable dissipative dispersive non linear wave system are investigated in terms of a nonlinear lattice model it is found in a three pulse periodic system in which either periodic or chaotic motions which are consistent with the results of the numerical initial value problem occur when the individual interacting pulse has an asymmetric oscillatory tail structure the existence of oscillatory structure is found to be responsible for chaotic motions several general properties of the asymmetric lattice are also discussed in comparison with the symmetric lattices for the korteweg de vries kdv and the fifth order kdv solitons

positive solutions for neumann elliptic problems involving critical hardy sobolev exponent with boundary singularities abstract in this paper we consider the existence of positive solution for some semilinear elliptic equations with neumann boundary condition involving a critical hardy sobolev exponent and hardy terms with boundary singularities using mountain pass lemma without ps condition and the strong maximum principle we get the existence of a positive solution

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

hardy sobolev critical singular elliptic equations with mixed dirichlet neumann boundary conditions abstract three important inequalities the poincare hardy and generalized poincare inequalities on the mixed boundary conditions are firstly established by some analytical techniques then the existence and multiplicity of positive solutions are studied for a class of semilinear elliptic equations with mixed dirichlet neumann boundary conditions involving hardy terms and hardy sobolev critical exponents by using the variational methods

a note on the sign changing solutions to elliptic problems with critical sobolev and hardy terms let rn be a bounded domain such that 0 n 7 2 2nn 2 we obtain existence of sign changing solutions for the dirichlet problem u u x 2 u 2 2u u on u 0 on for suitable positive numbers and

solutions for a singular critical growth problem with a weight in this paper we consider some semilinear elliptic equations with hardy potential by using linking theorem in p rabinowitz minimax methods in critical points theory with applications to differential equations cbms reg conf ser math vol 65 amer math soc providence ri 1986 and analyzing the effect of nonlinearities we establish the existence of nontrivial solutions

minimizers of caffarelli kohn nirenberg inequalities with the singularity on the boundary let be a bounded smooth domain in bf r n n geqq 3 and d a 1 2 omega be the completion of c 0 infty omega with respect to the norm u a 2 int omega x 2a nabla u 2 d x the caffarelli kohn nirenberg inequalities state that there is a constant c 0 such that begin array ll int omega x bq u q d x frac 2 q leqq c int omega x 2a nabla u 2 d x end array quad quad 0 1 for u in d a 1 2 omega and infty a frac n 2 2 quad 0 leqq b a leqq 1 quad q frac 2n n 2 2 b a we prove the best constant for 0 1 s a b omega inf limits u in d a 1 2 backslash 0 frac int omega x 2a nabla u 2 d x int omega x bq u q d x frac 2 q is always achieved in d a 1 2 omega provided that 0 in partial omega and the mean curvature h 0 0 where a b satisfies i a 0 rm and n geqq 4 if a 0 and 1 b 0 then the result was proved by ghoussoub and robert 12

multiple positive solutions for robin problem involving critical weighted hardy sobolev exponents with boundary singularities abstract this paper deals with the existence of positive solutions for robin elliptic problems involving critical weighted hardy sobolev exponents with boundary singularities using the caffarelli kohn nirenberg inequalities and variational methods we prove the existence and multiplicity of positive solutions

existence and multiplicity of symmetric solutions for semilinear elliptic equations with singular potentials and critical hardy sobolev exponents abstract this paper deals with the singular semilinear elliptic problem div x 2 a u u x 2 1 a q x u p 2 u x b p h x u in u 0 on where r n n 3 is a smooth bounded domain 0 and is g symmetric with respect to a subgroup g of o n 0 a n 2 2 0 0 with n 2 2 a 2 2 a b a 1 p p a b 2 n n 2 1 a b q x is continuous and g symmetric on and h r r is a continuous nonlinearity of lower order satisfying some conditions based upon the symmetric criticality principle of palais and variational methods we prove several existence and multiplicity results of g symmetric solutions under certain appropriate hypotheses on q and h

existence and multiplicity of positive solutions for elliptic equation with critical weighted hardy sobolev exponents and boundary singularities abstract we study a semilinear elliptic equation involving critical weighted hardy sobolev exponents with boundary singularities the existence and multiplicity of positive solutions are established our method relies upon ekeland s variational principle and mountain pass lemma

nonuniqueness of solutions for a class of forward backward parabolic equations abstract we study the initial boundary value problem u t u x x e u t x x in 0 u e u t 0 in 0 u u 0 0 in 0 with measure valued initial data here is a bounded open interval 0 0 is increasing in 0 and decreasing in and the regularising term is increasing but bounded it is natural to study measure valued solutions since singularities may appear spontaneously in finite time nonnegative radon measure valued solutions are known to exist and their construction is based on an approximation procedure until now nothing was known about their uniqueness in this note we construct some nontrivial examples of solutions which do not satisfy all properties of the constructed solutions whence uniqueness fails in addition we classify the steady state solutions

solutions for semilinear elliptic equations with critical exponents and hardy potential abstract in this paper we answer affirmatively an open problem cf theorem 4 in ferrero and gazzola j differential equations 177 2001 494 let 0 be an open bounded domain r n n 5 and assume that 0 n 2 2 2 n 2 n 2 then for all 0 there exists a nontrivial solution with critical level in the range 0 1 n s n 2 for the problem u u x 2 u u 2 2 u in u 0 on

analysis of zero modes for dirac operators with magnetic links abstract in this paper we provide a means to approximate dirac operators with magnetic fields supported on links in s 3 and r 3 by dirac operators with smooth magnetic fields then we proceed to prove that under certain assumptions the spectral flow of paths along these operators is the same in both the smooth and the singular case we recently characterized the spectral flow of such paths in the singular case this allows us to show the existence of new smooth compactly supported magnetic fields in r 3 for which the associated dirac operator has a non trivial kernel using clifford analysis we also obtain criteria on the magnetic link for the non existence of zero modes

existence and multiplicity of positive solutions for some hardy sobolev critical elliptic equation with boundary singularities abstract in this paper by ekeland s variational principle and strong maximum principle we consider the existence and multiplicity of positive solutions for some semilinear elliptic equation involving critical hardy sobolev exponents and hardy terms with boundary singularities

on a pseudoparabolic regularization of a forward backward forward equation abstract we consider an initial boundary value problem for a degenerate pseudoparabolic regularization of a nonlinear forward backward forward parabolic equation with a bounded nonlinearity which is increasing at infinity we prove existence of suitably defined nonnegative solutions of the problem in a space of radon measures solutions satisfy several monotonicity and regularization properties in particular their singular part is nonincreasing and may disappear in finite time the problem is of intrinsic mathematical interest but also arises naturally when studying by time reversal the spontaneous appearance of singularities in a specific application

the effect of domain topology on the number of positive solutions for singular elliptic problems involving the caffarelli kohn nirenberg inequalities abstract in this paper we consider the critical singular equation involving the caffarelli kohn nirenberg inequalities of the type p 1 div x 2 a u u x 2 1 a u q 2 u x d d u p 2 u x b p x u 0 x u 0 x here is a bounded domain with smooth boundary in r n and contains 0 in its interior 0 a 2 n 2 2 2 n 3 a b a 1 a d a 1 p p a b 2 n n 2 1 a b d d a d 2 n n 2 1 a d is a positive parameter and 2 q d by lusternik schnirelmann category theory we prove that the problem p 1 has at least cat positive solutions

multiple positive solutions for semilinear elliptic equations with critical weighted hardy sobolev exponents abstract the existence and multiplicity of positive solutions are obtained for a class of semilinear elliptic equations with critical weighted hardy sobolev exponents and the concave convex nonlinearity by variational methods and some analysis techniques

existence and multiplicity of positive solutions for a perturbed semilinear elliptic equation with two hardy sobolev critical exponents abstract in this paper by applying ekeland s variational principle and strong maximum principle we investigate the existence and multiplicity of positive solutions for a perturbed semilinear elliptic equation with two critical hardy sobolev exponents and boundary singularities

quasilinear elliptic problems with critical exponents and hardy terms abstract let 0 be an open bounded domain r n n p 2 we are concerned with the multiplicity of positive solutions of p u u p 2 u x p u p 2 u q x u p 2 u u w 0 1 p where p u div u p 2 u 1 p n p np n p 0 n p p p 0 and q x is a nonnegative function on by investigating the effect of the coefficient of the critical nonlinearity we by means of variational method prove the existence of multiple positive solutions

positive solutions to the weighted critical quasilinear problems in a bounded domain with smooth boundary we consider a kind of weighted quasilinear elliptic problem which satisfies dirichlet boundary condition and involves the hardy sobolev inequality by the analytic techniques we first get the properties of the extremal functions by which the best hardy sobolev constant is achieved then by the variational methods the existence of positive solutions to the problem is verified by careful estimates and computations

existence and multiplicity of symmetric solutions for the weighted critical quasilinear problems this paper deals with the weighted critical quasilinear problem div u p 2 u x a p m u p 2u x p a 1 q x u p a b 2u x b p a b h x u in w u 0on w where w r n is a smooth bounded domain 0 w and w is g symmetric with respect to a subgroup g of o n 1

existence and multiplicity of positive solutions for a class of semilinear elliptic equations involving hardy term and hardy sobolev critical exponents abstract existence and multiplicity of positive solutions are studied for a class of semilinear elliptic equations with hardy term hardy sobolev critical exponents and sublinear nonlinearity by variational methods and some analysis techniques

existence and multiplicity of solutions for semilinear elliptic equations with critical weighted hardy sobolev exponents abstract some existence results are obtained for solutions of semilinear elliptic equations with critical weighted hardy sobolev exponents and superlinear nonlinearity by the variational methods and some analysis techniques

intrinsic stochasticity with many degrees of freedom we study a simple model equation describing a system with an infinity of degrees of freedom which displays an intrinsically chaotic behavior some concepts of fully developed turbulence are discussed in relation to this model we also develop an approach based on lyapunov exponent measurements numerical results on the distribution of lyapunov numbers and the power spectrum of the associated lyapunov vectors are presented and briefly discussed