existence of primitive divisors of lucas and lehmer numbers we prove that for n 30 every n th lucas and lehmer number has a primitive divisor this allows us to list all lucas and lehmer numbers without a primitive divisor

existence and uniqueness of multi bump bound states of nonlinear schrodinger equations with electromagnetic fields abstract in this paper we are concerned with the existence and uniqueness of multi bump bound states of the nonlinear schrodinger equations with electromagnetic potential i t i a x 2 v x p 2 x r n for sufficiently small 0 where i is the imaginary unit 2 p 2 n n 2 for n 3 and 2 p for n 1 2 v x is a bounded real function on r n and a x a 1 x a 2 x a n x is such that a j x is a bounded real function on r n for j 1 2 n for any finite collection of a 1 a 2 a k of non degenerate critical points of v x we show that there exists a solution x t exp iet u x which is a small perturbation of a sum of one bump solutions concentrated at a 1 a 2 a k and u x is unique up to rotations for small 0

block transitive automorphism groups of designs in this note we consider subgroups of the automorphism group of a 2 v k 1 design the notation is standard see for example 2 the number of blocks through a point is r v l k 1 and the number of blocks b is vr k recall that a flag is an incident point block pair there are two main results in this note the first is straightforward but essential to the proof of the second

radial solutions for the brezis nirenberg problem involving large nonlinearities abstract let us consider the problem 0 1 u a x u u p in b 1 u 0 in b 1 u 0 on b 1 where b 1 is the unit ball in r n n 3 and a x 0 is a smooth radial function under some suitable assumptions on the regular part of the green function of the operator u n 1 r u a r u we prove the existence of a radial solution to 0 1 for p large enough

on effective computation of fundamental units ii introduction in part i of this paper we described the theoretical improvements of our method of computing fundamental units in algebraic number fields which we achieved in the last few years those improvements were of considerable influence on the corresponding computer program since its first 1976 2 implementation it was completely rewritten and changed in so many details that a new presentation cannot be avoided in sections 1 and 2 of this paper we describe the applications of part i 6 to the algorithm for constructing fundamental units section 3 contains a complete list of numerical examples concerning algebraic number fields of small degree and small absolute discriminant besides the fundamental units the tables contain much information about fields of degree five and six which was so far unknown the determination of those fields and their subfields is described in 3 besides the fundamental units we also listed the order of the torsion subgroup tuf of the unit group u whenever it is different from 2 it was computed by the methods of section 2 of 5 all computations were carried out on the control data cyber 76 of the computer center of the university of cologne

the diophantine equation nx 2 2 m y n let n be an odd integer with n3 in this paper using a recent result on the existence of primitive divisors of lehmer numbers given by y bilu g hanrot and p m voutier we prove that the equation nx 2 2 m y n has no positive integer solution x y m satisfying gcd x y 1 and m is odd

the first sign change of fourier coefficients of cusp forms let f sum no a n q n a n real be an arbitrary nonzero cusp form of even integral weight k geq 2 on gamma 0 n square free n we obtain an effective bound that the first sign change of a n occurs

bubble tower radial solutions in the slightly supercritical brezis nirenberg problem abstract in this paper we consider the brezis nirenberg problem in dimension n 4 in the supercritical case we prove that if the exponent gets close to n 2 n 2 and if simultaneously the bifurcation parameter tends to zero at the appropriate rate then there are radial solutions which behave like a superposition of bubbles namely solutions of the form j 1 k 1 1 m j 4 n 2 y 2 n 2 2 m j 1 o 1 n n 2 n 2 4 where mj and mj o mj 1 for all j these solutions lie close to turning points to the right of the associated bifurcation diagram

peak solutions for the fractional nirenberg problem abstract we consider the fractional nirenberg problem u k u n 2 n 2 u 0 in r n assume k x 1 e k x and k x has at least two critical points satisfying some local conditions the existence of two peak solutions is obtained when the positive number e is small enough

blowup analysis for a nonlinear equation with negative exponent abstract in this paper we study the blowup behavior of the minimizing sequence to certain nonlinear elliptic equation with negative critical exponents we analyze the location for blowup point and the asymptotic behavior of solutions near the blowup point

on the number of solutions of the generalized ramanujan nagell equation let d1 and d2 be coprime positive integers and let k be an odd positive integer coprime with d1d2 we consider the diophantine equation d1x2 d2 kn in the unknowns x 1 n 1 we give a necessary and sufficient condition on d1 d2 and k under which this equation has at most 2 k 1 solutions where k denoted the number of distinct prime divisors of k thus under a necessary and sufficient conditon the equation has at most one solution whenever k is a prime we also consider some related equations and we prove that the diophantine equation x2 7 4yn has no solution in integers x 1 y 2 and n 1

type ii ancient compact solutions to the yamabe flow nsf 0604657 1266172 0905749 1056387 fondo basal cmm millenium nucleus capde nc130017 fondecyt 1150066

the lazer mckenna conjecture for an elliptic problem with critical growth part ii abstract we consider an elliptic problem of ambrosetti prodi type involving critical sobolev exponent we prove that this problem has solutions blowing up near the boundary of the domain

positive solutions of elliptic equations involving supercritical growth positive radial solutions of elliptic equation involving supercritical growth are analysed as their supremum norm tends to infinity it is shown that they converge uniformly away from the origin as well as in h 1 to the unique singular solution

qualitative properties of multi bubble solutions for nonlinear elliptic equations involving critical exponents abstract the objective of this paper is to obtain qualitative characteristics of multi bubble solutions to the lane emden fowler equations with slightly subcritical exponents given any dimension n 3 by examining the linearized problem at each m bubble solution we provide a number of estimates on the first n 2 m eigenvalues and their corresponding eigenfunctions specifically we present a new and unified proof of the classical theorems due to bahri li rey 1995 2 and rey 1999 24 which state that if n 4 or n 3 respectively then the morse index of a multi bubble solution is governed by a certain symmetric matrix whose component consists of a combination of green s function the robin function and their first and second derivatives

primitive permutation groups of odd degree and an application to finite projective planes one of the most beautiful and important results concerning finite projective planes is the ostrom wagner theorem 26 such a plane admitting a 2 transitive collineation group must be desarguesian it has long been conjectured that the same conclusion must hold if it is only assumed that there is a collineation group transitive on incident point line pairs ll pp 208 214 171 the starting point for this paper was a proof of this conjecture modulo a degenerate situation

the toda system and clustering interfaces in the allen cahn equation we consider the allen cahn equation varepsilon 2 delta u 1 u 2 u 0 in a bounded smooth domain in mathbb r 2 under zero neumann boundary conditions where varepsilon 0 is a small parameter let 0 be a segment contained in connecting orthogonally the boundary under certain nondegeneracy and nonminimality assumptions for 0 satisfied for instance by the short axis in an ellipse we construct for any given n 1 a solution exhibiting n transition layers whose mutual distances are o varepsilon log varepsilon and which collapse onto 0 as varepsilon to 0 asymptotic location of these interfaces is governed by a toda type system and yields in the limit broken lines with an angle at a common height and at main order cutting orthogonally the boundary

existence of sign changing solutions for an equation with a weighted p laplace operator we consider radial solutions of a general elliptic equation involving a weighted p laplace operator with a subcritical nonlinearity by a shooting method we prove the existence of solutions with any prescribed number of nodes the method is based on a change of variables in the phase plane a very general computation of an angular velocity and new estimates for the decay of an energy associated with an asymptotic hamiltonian problem estimating the rate of decay for the energy requires a sub criticality condition the method covers the case of solutions which are not compactly supported or which have compact support in the last case we show that the size of the support increases with the number of nodes

periodicity of signs of fourier coefficients of eta quotients we study the periodicity of signs of fourier coefficients of the function d f qd rd where is a positive integer f q n 1 1 qn and rd z as an application we will prove the periodicity of signs for the crank differences conjectured by g e andrews and r lewis and for the weighted counts of certain types of partitions

an improved toeplitz algorithm for polynomial matrix null space computation in this paper we present an improved algorithm to compute the minimal null space basis of polynomial matrices a problem which has many applications in control and systems theory this algorithm takes advantage of the block toeplitz structure of the sylvester matrix associated with the polynomial matrix the analysis of algorithmic complexity and numerical stability shows that the algorithm is reliable and can be considered as an efficient alternative to the well known pencil state space algorithms found in the literature

multi peak solutions for super critical elliptic problems in domains with small holes abstract this paper deals with the slightly super critical elliptic problem u u n 2 n 2 e in u 0 in u 0 on 1 where e 0 is a small parameter and r n is a bounded domain with smooth boundary assuming that the domain exhibits k sufficiently small holes multiple solutions are constructed by gluing double spike patterns located near each of the holes

the lin ni conjecture in negative geometries abstract we prove see theorem 1 1 that the lin ni conjecture for closed manifolds is false in dimensions n 4 5 when the scalar curvature is negative but that it holds true in the bounded energy setting when n 6 see theorem 1 2 as a corollary to theorem 1 1 we prove see theorem 2 2 that contrary to the 3 dimensional case the sup inf inequality does not hold in general with a sole control on the size of the potential for positive solutions of stationary critical schrodinger equations

the role of the green s function in a non linear elliptic equation involving the critical sobolev exponent this paper is concerned with non linear elliptic problems of the type pe u u n 2 n 2 eu u 0 on u 0 on where is a smooth and bounded domain in rn n 4 and e 0 we show that if the ue are solutions of pe which concentrate around a point as e 0 then this point cannot be on the boundary of and is a critical point of the regular part of the green s function conversely we show that for n 5 and any non degenerate critical point x0 of the regular part of the green s function there exist solutions of pe concentrating around x0 as e 0

on the first sign change of hecke eigenvalues of newforms let f be a non zero cusp form of even integral weight k 2 and level n with real fourier coefficients a n n n using a classical theorem of landau together with the facts that the hecke l function of f is entire and the rankin selberg zeta function of f has a pole at its abscissa of convergence with residue essentially equal to the square of the petersson norm of f one easily shows that there are infinitely many n n such that a n 0 as well as infinitely many n with a n 1 siegel s argument does not apply and the result above is not known now suppose that n is squarefree and f is a newform which is a normalized hecke eigenform so a 1 1 and a n is the n th hecke eigenvalue of f we shall prove

on some first sign change problems of modular forms abstract we give two different bounds namely for the first prime p such that the p th hecke eigenvalue f p is negative when f is a normalized cuspidal newform of level 0 n and for the first sign change for general cusp forms which improves the bound given by choie and kohnen 3 in the level aspect in the case of prime level

blow up phenomena for the yamabe equation it is well known that the pde 1 has at least one positive solution for any choice of m g if n 6 and m g is not locally conformally flat this follows from results of t aubin 3 the remaining cases were solved by r schoen using the positive mass theorem 16 solutions to 1 are not usually unique as an example consider the product metric on 51 l x 5n 1 l if l is sufficiently small then the yamabe pde has a unique solution on the other hand there are many non minimizing solutions if l is large d pollack 14 has used gluing techniques to construct high energy solutions on more general background manifolds given any conformai class with positive yamabe constant and any positive integer iv there exists a new conformai class which is close to the original one in the c norm and contains at least n metrics of constant scalar curvature see 14 theorem 0 1 it is an interesting question whether the set of all solutions to the yamabe pde is compact in the c2 topology say a well known conjecture states that this should be true unless m p is conformally equivalent to the round sphere see 17 18 19 this conjecture has been verified in low dimensions and in the locally conformally flat case if m g is locally conformally flat compactness follows from work of r schoen 17 18 moreover schoen proposed a strategy for proving the conjecture in the non locally conformally flat case based on the pohozaev identity

classification of low energy sign changing solutions of an almost critical problem in this paper we make the analysis of the blow up of low energy sign changing solutions of a semilinear elliptic problem involving nearly critical exponent our results allow to classify these solutions according to the concentration speeds of the positive and negative part and in high dimensions lead to complete classification of them additional qualitative results such as symmetry or location of the concentration points are obtained when the domain is a ball

geometry of phase space and solutions of semilinear elliptic equations in a ball we consider the problem u u p u in in b u 0 in b u 0 on b where b denotes the unit ball in r n n 3 a 0 and p 1 merle and peletier showed that for p n 2 n 2 there is a unique value a 0 such that a radial singular solution exists this value is the only one at which an unbounded sequence of classical solutions of 1 may accumulate here we prove that if additionally p 0 0 in b u 0 on b where f s s p s q 1 q p and p satisfies the same condition as above

phase plane analysis and classification of solutions of a mixed sublinear superlinear elliptic problem abstract in this paper some mixed sublinear superlinear critical problem extending the famous problem of brezis nirenberg are analysed the existence of solutions is discussed a phase plane analysis is performed in order to transform the problem into an ordinary differential equation finally a full classification of radial solutions according to their behavior at the origin is performed for subcritical critical and supercritical cases

multiple positive solutions for a schrodinger poisson slater system abstract in this paper we investigate the existence of positive solutions to the following schrodinger poisson slater system u u u u p 2 u in u 2 in u 0 on where is a bounded domain in r 3 is a fixed positive parameter and p 2 2 n n 2 we prove that if p is near the critical sobolev exponent 2 then the number of positive solutions is greater then the lusternik schnirelmann category of

on divisors of lucas and lehmer numbers let un be the nth term of a lucas sequence or a lehmer sequence in this article we shall establish an estimate from below for the greatest prime factor of un which is of the form n exp log n 104 log log n in doing so we are able to resolve a question of schinzel from 1962 and a conjecture of erd s from 1965 in addition we are able to give the first general improvement on results of bang from 1886 and carmichael from 1912

primitive divisors of lucas and lehmer sequences stewart reduced the problem of determining all lucas and lehmer sequences whose nth element does not have a primitive divisor to solving certain thue equations using the method of tzanakis and de weger for solving thue equations we determine such sequences for n 30 further computations lead us to conjecture that for n 30 the nth element of such sequences always has a primitive divisor

a special class of stationary flows for two dimensional euler equations a statistical mechanics description we consider the canonical gibbs measure associated to an vortex system in a bounded domain at inverse temperature widetilde beta and prove that in the limitn widetilde beta n n 1 where 8 here denotes the vorticity intensity of each vortex the one particle distribution function n nx x converges to a superposition of solutions of the following mean field equation left begin array 20 c varrho beta x frac e beta psi mathop smallint limits lambda e beta psi delta psi varrho beta in lambda psi partial lambda 0 end array right

boundary towers of layers for some supercritical problems we consider the supercritical problem u u p 1uin d u 0on d where d is a bounded smooth domain in rn and p is smaller than the th critical sobolev exponent 2n n 2n 2 with 1 n 3 we show that in some suitable torus like domains d there exists an arbitrary large number of sign changing solutions with alternate positive and negative layers which concentrate at different rates along a dimensional submanifold of d as p approaches 2n from below

coefficient of maass forms and the siegel zero with an appendix by dorian goldfeld jeffrey hoffstein and daniel lieman

a short remark regarding pohozaev type results on general domains assuming finite morse index we are interested in nonnegative nontrivial solutions of u up in u 0 on 1 where 1 p and a bounded smooth domain in rn with 3 n 9 we show that given a nonnegative integer m there is some large p m such that the only nonnegative solution u of morse index at most m is u 0 2010 mathematics subject classification

hardy sobolev equations with asymptotically vanishing singularity blow up analysis for the minimal energy abstract we study the asymptotic behavior of a sequence of positive solutions u 0 as 0 to the family of equations u a x u u 2 s 1 x s in u 0 on where s 0 is a sequence of positive real numbers such that lim 0 s 0 2 s 2 n s n 2 and r n is a bounded smooth domain such that 0 when the sequence u 0 is uniformly bounded in l then up to a subsequence it converges strongly to a minimizing solution of the stationary schrodinger equation with critical growth in case the sequence blows up we obtain strong pointwise control on the blow up sequence and then using the pohozaev identity localize the point of singularity which in this case can at most be one and derive precise blow up rates in particular when n 3 or a 0 then blow up can occur only at an interior point of or the point 0

asymptotic behaviour of positive solutions of elliptic equations with critical and supercritical growth ii the nonradial case abstract we consider positive solutions ue of a family of elliptic boundary value problems ie as e 0 they approach in some sense the problem i0 which has a nonlinearity with critical or supercritical growth it is shown how the solution ue concentrates at a point how this happens asymptotically and where in the domain it happens

sign changing bubble towers for asymptotically critical elliptic equations on riemannian manifolds given a smooth compact riemannian n manifold m g we consider the equation gu hu u 2 2 eu where h is a c1 function on m the exponent 2 2n n 2 is the critical sobolev exponent and e is a small positive real parameter such that e 0 we prove the existence of blowing up families of sign changing solutions which develop bubble towers at some point where the function h is greater than the yamabe potential n 24 n 1 scalg

on a fourth order elliptic equation involving the critical sobolev exponent the effect of the graph topology abstract in this paper we consider the following nonlinear equation 2 u k u n 4 n 4 e u with u 0 in and u u 0 on where is a smooth bounded domain in r n n 8 k is a smooth positive function in and e is a small positive parameter using the effect of the graph topology of the function k we prove existence multiplicity and concentration of solutions we prove also a nonexistence result

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

bubble towers for supercritical semilinear elliptic equations we construct positive solutions of the semilinear elliptic problem u u up 0 with dirichet boundary conditions in a bounded smooth domain rn n 4 when the exponent p is supercritical and close enough to n 2n 2 and the parameter r is small enough as p n 2n 2 the solutions have multiple blow up at finitely many points which are the critical points of a function whose definition involves green s function our result extends the result of del pino et al j differential equations 193 2 2003 280 when is a ball and the solutions are radially symmetric

lazer mckenna conjecture the critical case abstract we consider an elliptic problem of ambrosetti prodi type involving critical sobolev exponent on a bounded smooth domain of dimension six or higher by constructing solutions with many sharp peaks near the boundary of the domain but not on the boundary we prove that the number of solutions for this problem is unbounded as the parameter tends to infinity thereby proving the lazer mckenna conjecture in the critical case

infinitely many solutions for the nonlinear schrodinger equations with magnetic potentials in rn in this paper we study a nonlinear schrodinger equations with magnetic potentials in rn involving subcritical growth under some decaying and weak symmetry conditions of both electric and magnetic potentials we prove that the equation has infinitely many nonradial complex valued solutions by applying the finite reduction method

blow up and symmetry of sign changing solutions to some critical elliptic equations abstract in this paper we continue the analysis of the blow up of low energy sign changing solutions of semi linear elliptic equations with critical sobolev exponent started in m ben ayed k el mehdi f pacella blow up and nonexistence of sign changing solutions to the brezis nirenberg problem in dimension three ann inst h poincare anal non lineaire in press in addition we prove axial symmetry results for the same kind of solutions in a ball

critical points of the regular part of the harmonic green function with robin boundary condition abstract in this paper we consider the green function for the laplacian in a smooth bounded domain r n with robin boundary condition g b x g 0 on and its regular part s x y where b 0 is smooth we show that in general as the robin function r x s x x has at least 3 critical points moreover in the case b const we prove that r has critical points near non degenerate critical points of the mean curvature of the boundary and when b const there are critical points of r near non degenerate critical points of b

a generalized cubic moment and the petersson formula for newforms using a cubic moment we prove a weyl type subconvexity bound for the quadratic twists of a holomorphic newform of square free level trivial nebentypus and arbitrary even weight this generalizes work of conrey and iwaniec in that the newform that is being twisted may have arbitrary square free level and also that the quadratic character may have even conductor one of the new tools developed in this paper is a more general petersson formula for newforms of square free level

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

oscillations of fourier coefficients of modular forms a p 2p cos 0 p since we know the truth of the ramanujan petersson conjecture it follows that the 0 p s are real inspired by the sato tate conjecture for elliptic curves serre 14 conjectured that the 0 p s are uniformly distributed in the interval 0 rc with respect to the 1 measure sin2odo following serre we shall refer to this as the sato tate r conjecture there being no room for confusion it has been known for a long time that the truth of this conjecture implies much about the oscillatory behaviour of the fourier coefficients in particular the following is implied by the sato tate conjecture theorem 1 for any normalized hecke eigenform

boundary bubbling solutions for a supercritical neumann problem with mixed nonlinearities abstract in this paper we investigate the existence of boundary bubbling solutions for a supercritical neumann problem with mixed nonlinearities under a suitable assumption on the mean curvature of the boundary the behavior of the solutions is as a tower of k bubbles when the exponent goes to sobolev critical exponent and the blow up point is a critical point of the mean curvature

nodal solutions for an elliptic problem involving large nonlinearities abstract let us consider the problem 0 1 u a x u u p 1 u in b 1 u 0 on b 1 where b 1 is the unit ball in r n n 3 and a x 0 is a smooth radial function under some suitable assumptions on the regular part of the green function of the operator u n 1 r u a r u we prove the existence of a changing sign solution to 0 1 for p large enough

segregated and synchronized vector solutions for nonlinear schrodinger systems we consider the following nonlinear schrodinger system in mathbb r 3 left begin array ll delta u p x u mu u 2 u beta v 2u quad x in mathbb r 3 delta v q x v nu v 2 v beta u 2v quad x in mathbb r 3 end array right where p r and q r are positive radial potentials mu 0 nu 0 and beta in mathbb r is a coupling constant this type of system arises in particular in models in bose einstein condensates theory

class number computations of real abelian number fields in this paper we describe the calculation of the class numbers of most real abelian number fields of conductor 200 the technique is due to j m masley and makes use of discriminant bounds of a m odlyzko in several cases we have to assume the generalized riemann hypothesis introduction it is well known that the class number h of an abelian number field can be written as h h h where h is the class number of the maximal real subfield k of k and h is an integer we can determine the relative class number h in a straightforward way using the complex analytic class number formula see 7 kap iii or 9 chapter 3 section 3 for the full cyclotomic fields q n with 4o n 256 and their subfields one can deduce hfrom the tables of g schrutka von rechtenstamm 15 here dn denotes a primitive nth root of unity and 40 is the euler function for the class number factor h the complex analytic class number formula is less useful since it requires that the units of k be known alternative techniques have been developed by j m masley 13 who computed the class number of almost all real cyclic number fields of conductor 100 here the conductor of k is the least f for which k c q tf in this paper we apply masley s techniques with a few additions to determine the class numbers of a large collection of real abelian fields of conductor 200 see section 1 for a precise statement of our results some of which assume the generalized riemann hypothesis an important ingredient of masley s method is the use of discriminant lower bounds proved by a m odlyzko 14 these lead to an upper bound for the class number of a real abelian number field provided that its conductor or more precisely its root discriminant see 13 section 1 is sufficiently small it follows that this method can only be used for a finite number of real abelian number fields the existence of infinite class field towers shows that this remains true after any future improvement of odlyzko s bounds in fact examples of j martinet 12 show that the method will never apply to fields whose root discriminant is larger than five times the present bound under assumption of the generalized riemann hypothesis the structure of this paper is as follows section 1 contains our results and section 2 lists the theorems used in the proofs the proofs themselves are largely suppressed received october 24 1980 revised december 24 1981 1980 mathematics subject classification primary 12 04 12a35 12a55

convergence of the yamabe flow for arbitrary initial energy we consider the yamabe flow frac partial g partial t r g r g g where g is a riemannian metric on a compact manifold m r g denotes its scalar curvature and r g denotes the mean value of the scalar curvature we prove convergence of the yamabe flow if the dimension n satisfies 3 leq n leq 5 or the initial metric is locally conformally flat

solving thue equations without the full unit group the main problem when solving a thue equation is the computation of the unit group of a certain number field it is shown that the knowledge of a subgroup of finite index is in fact sufficient two examples linked with the primitive divisor problem for lucas and lehmer sequences are given la principale difficulte lors de la resolution d une equation de thue reside dans le calcul du groupe des unites du corps de nombres associe on montre qu il suffit en fait de connaitre un sous groupe d indice fini de ce groupe on donne deux exemples li

finite linear spaces admitting a projective group psu 3 q with q even abstract this article is a contribution to the study of the automorphism groups of finite linear spaces in particular we look at simple groups and prove the following theorem let g psu 3 q with q even and g acts line transitively on a finite linear space s then s is one of the following cases i a projective plane ii a regular linear space with parameters b v r k q2 q2 q 1 q3 1 q2 q 1 q 1 this is called the hermitian unitary design

the effect of linear perturbations on the yamabe problem in conformal geometry the compactness conjecture asserts that the set of yamabe metrics on a smooth compact aspherical riemannian manifold left m g right is compact established in the locally conformally flat case by schoen lecture notes in mathematics vol 1365 pp 120 154 springer berlin 1989 surveys pure application and mathematics 52 longman science technology pp 311 320 harlow 1991 and for n le 24 by khuri marques schoen j differ geom 81 1 143 196 2009 it has revealed to be generally false for n ge 25 as shown by brendle j am math soc 21 4 951 979 2008 and brendle marques j differ geom 81 2 225 250 2009 a stronger version of it the compactness under perturbations of the yamabe equation is addressed here with respect to the linear geometric potential frac n 2 4 n 1 mathrm scal g mathrm scal g being the scalar curvature of left m g right we show that a priori l infty bounds fail for linear perturbations on all manifolds with n ge 4 as well as a priori gradient l 2 bounds fail for non locally conformally flat manifolds with n ge 6 and for locally conformally flat manifolds with n ge 7 in several situations the results are optimal our proof combines a finite dimensional reduction and the construction of a suitable ansatz for the solutions generated by a family of varying metrics in the conformal class of g

concentrating solutions for a planar elliptic problem involving nonlinearities with large exponent abstract we consider the boundary value problem u u p 0 in a bounded smooth domain in r 2 with homogeneous dirichlet boundary condition and p a large exponent we find topological conditions on which ensure the existence of a positive solution u p concentrating at exactly m points as p in particular for a nonsimply connected domain such a solution exists for any given m 1

asymptotic behavior of solutions for nonlinear elliptic problems with the fractional laplacian abstract in this paper we study the asymptotic behavior of least energy solutions and the existence of multiple bubbling solutions of nonlinear elliptic equations involving the fractional laplacians and the critical exponents this work can be seen as a nonlocal analog of the results of han 1991 24 and rey 1990 35

a prescribed scalar curvature type equation almost critical manifolds and multiple solutions we present an asymptotic analysis for a perturbed prescribed scalar curvature type equation a major consequence is a non existence result in low dimension conversely we prove an existence result in higher dimensions to this aim we develop a general finite dimensional reduction procedure for perturbed variational functionals the general principle can be useful to discuss some other nonlinear elliptic pde with sobolev critical growth in bounded domains

finite projective planes admitting a projective linear group psl 2 q abstract let s be a projective plane and let g aut s and psl 2 q g p l 2 q with q 3 if g acts point transitively on s then q 7 and s is of order 2

bubble tower solutions to asymptotically critical nonlocal elliptic equations on conformal infinities abstract assume that x g is an asymptotically hyperbolic manifold with conformal infinity m h we construct positive and sign changing bubble tower solutions to low order perturbations of yamabe equations 0 1 on m h these solutions exhibit isolated but non simple blow up in the sense of schoen 30