sobolev type embedding and quasilinear elliptic equations with radial potentials abstract we study the existence of nontrivial radial solutions for quasilinear elliptic equations with unbounded or decaying radial potentials the existence results are based upon several new embedding theorems we establish in the paper for radially symmetric functions

on a semilinear elliptic neumann problem with asymmetric nonlinearities abstract we consider the neumann problem 1 1 below we extend therange of applicability of the sharp nonresonance condition derived in go om so as in particular allow asymmetric nonlinearities 1 introduction this paper is mainly concerned with the neumann problem i au g u h x inq 1 1 du du 0 on oil here a is the laplacian operator q is a bounded domain of rn n 1 with c1 1 boundary and unit exterior normal v g is a continuous function on r and h is taken in l f2 or in lp q in a previous work go om we showed that if g s 2g s 1 2 lim sup oo s s oo s or s oo then a necessary and sufficient condition for 1 1 to be solvable for any h l f2 is that the range of g be all of k here kx 0 k2 denote the distinct eigenvalues of a on q with neumann boundary condition andg s jo g t dt the surjectivity of g thus provides a nonresonance condi tion at the eigenvalue xx 0 we refer to go om for historical background

the beginning of the fucik spectrum for the p laplacian has a nontrivial solution u here 1 p 2p u div u p 2 u u max u 0 u uu in 1 1 denoting by 1 2 the first two eigenvalues of 2p on w 1 p 0 0 cf the end of this introduction 7p clearly contains in particular 1 1 2 2 and the two lines 1 r and r 1 this generalized notion of spectrum was introduced for p 2 i e in the linear case in the 1970s by fuc ik 21 and dancer 15 in connection article id jdeq 1999 3645 available online at http www idealibrary com on

eigenvalues outside the bulk of inhomogeneous erdős renyi random graphs in this article an inhomogeneous erd s renyi random graph on 1 ldots n is considered where an edge is placed between vertices i and j with probability varepsilon n f i n j n for i le j the choice being made independently for each pair the integral operator i f associated with the bounded function f is assumed to be symmetric non negative definite and of finite rank k we study the edge of the spectrum of the adjacency matrix of such an inhomogeneous erd s renyi random graph under the assumption that n varepsilon n rightarrow infty sufficiently fast although the bulk of the spectrum of the adjacency matrix scaled by sqrt n varepsilon n is compactly supported the kth largest eigenvalue goes to infinity it turns out that the largest eigenvalue after appropriate scaling and centering converges to a gaussian law if the largest eigenvalue of i f has multiplicity 1 if i f has k distinct non zero eigenvalues then the joint distribution of the k largest eigenvalues converge jointly to a multivariate gaussian law the first order behaviour of the eigenvectors is derived as a byproduct of the above results the results complement the homogeneous case derived by 18

universality in several matrix models via approximate transport maps we construct approximate transport maps for perturbative several matrix models as a consequence we deduce that local statistics have the same asymptotic as in the case of independent gue or goe matrices i e they are given by the sine kernel in the bulk and the tracy widom distribution at the edge and we show averaged energy universality i e universality for averages of m points correlation functions around some energy level e in the bulk as a corollary these results yield universality for self adjoint polynomials in several independent gue or goe matrices which are close to the identity

a neumann problem with asymmetric nonlinearity and a related minimizing problem abstract variational methods are used to prove the existence of solution of some classes of nonlinear ordinary differential equations with zero neumann boundary conditions among other very mild conditions the nonlinearities we consider have a behavior at at least like a linearity of slope less than 2 4 a variational characterization of this valuevis used to prove both geometrical and compactness properties of the associated euler functional a partial differential equation version of these problems is also considered

limit laws for random matrices and free products in earlier articles we studied a kind of probability theory in the framework of operator algebras with the tensor product replaced by the free product we prove here that free random variables naturally arise as limits of random matrices and that wigner s semicircle law is a consequence of the central limit theorem for free random variables in this way we obtain a non commutative limit distribution of a general gaussian random matrix as an operator in a certain operator algebra wigner s law being given by the trace of the spectral measure of the selfadjoint component of this operator

quasilinear elliptic equations on noncompact riemannian manifolds abstract the existence of solutions to a class of quasilinear elliptic problems on noncompact riemannian manifolds with finite volume is investigated boundary value problems with homogeneous neumann conditions in possibly irregular euclidean domains are included as a special instance a nontrivial solution is shown to exist under an unconventional growth condition on the right hand side which depends on the geometry of the underlying manifold the identification of the critical growth is a crucial step in our analysis and entails the use of the isocapacitary function of the manifold a condition involving its isoperimetric function is also provided

the pearcey process the extended airy kernel describes the space time correlation functions for the airy process which is the limiting process for a polynuclear growth model the airy functions themselves are given by integrals in which the exponents have a cubic singularity arising from the coalescence of two saddle points in an asymptotic analysis pearcey functions are given by integrals in which the exponents have a quartic singularity arising from the coalescence of three saddle points a corresponding pearcey kernel appears in a random matrix model and a brownian motion model for a fixed time this paper derives an extended pearcey kernel by scaling the brownian motion model at several times and a system of partial differential equations whose solution determines associated distribution functions we expect there to be a limiting nonstationary process consisting of infinitely many paths which we call the pearcey process whose space time correlation functions are expressible in terms of this extended kernel

nontrivial solutions for a class of resonant p laplacian neumann problems we consider a nonlinear neumann problem driven by the p laplacian differential operator with a caratheodory reaction term we assume that asymptotically at infinity resonance occurs with respect to the principal eigenvalue 0 0 i e the reaction term is p 1 sublinear near using variational methods based on the critical point theory and an alternative minimax characterization of the first nonzero eigenvalue 1 0 we show that the problem has a nontrivial smooth strong solution

c1 α local regularity of weak solutions of degenerate elliptic equations abstract it is demonstrated the local c 1 alpha nature of weak solutions of elliptic equations of the type 1 1 in the introduction under the degeneracy or singularity assumptions a sub 1 a sub 3

non radial solutions for a non homogeneous henon equation abstract in this paper we study a henon like equation see eq 1 where the nonlinearity f t is not homogeneous i e it is not a power by minimization on the nehari manifold we prove that for large values of the parameter there is a breaking of symmetry and non radial solutions appear this holds for sub and super critical growth of the nonlinearity f

on universality of local edge regime for the deformed gaussian unitary ensemble we consider the deformed gaussian ensemble h n h n 0 m n in which h n 0 is a hermitian matrix possibly random and m n is the gaussian unitary random matrix gue independent of h n 0 assuming that the normalized counting measure of h n 0 converges weakly in probability if random to a non random measure n 0 with a bounded support and assuming some conditions on the convergence rate we prove the universality of the local eigenvalue statistics near the edge of the limiting spectrum of h n

the spectral norm of random inner product kernel matrices we study an inner product kernel random matrix model whose empirical spectral distribution was shown by xiuyuan cheng and amit singer to converge to a deterministic measure in the large n and p limit we provide an interpretation of this limit measure as the additive free convolution of a semicircle law and a marchenko pastur law by comparing the tracial moments of this matrix to an additive deformation of a wigner matrix we establish that for odd kernel functions the spectral norm of this matrix converges almost surely to the edge of the limiting spectrum our study is motivated by the analysis of a covariance thresholding procedure for the statistical detection and estimation of sparse principal components and our results characterize the limit of the largest eigenvalue of the thresholded sample covariance matrix in the null setting

existence and localization of solution for second order elliptic bvp in presence of lower and upper solutions without any order abstract we prove an existence and localization result for a solution of a second order elliptic equations with dirichlet and regular oblique derivative boundary conditions the main result is obtained assuming the existence of a lower and an upper solution without particular order and a positively homogeneous asymptotic behaviour of the right hand member of the equation

a nonresonance condition for radial solutions of a nonlinear neumann elliptic problem abstract we prove an existence result for radial solutions of a neumann elliptic problem whose nonlinearity asymptotically lies between the first two eigenvalues to this aim we introduce an alternative nonresonance condition with respect to the second eigenvalue which in the scalar case generalizes the classical one in the spirit of fonda et al 1991 2 our approach also applies for nonlinearities which do not necessarily satisfy a subcritical growth assumption

pdes for the gaussian ensemble with external source and the pearcey distribution the present paper studies a gaussian hermitian random matrix ensemble with external source given by a fixed diagonal matrix with two eigenvalues a as a first result the probability that the eigenvalues of the ensemble belong to an interval e satisfies a fourth order pde with quartic nonlinearity the variables are the eigenvalue a and the boundary of e this equation enables one to find a pde for the pearcey distribution the latter describes the statistics of the eigenvalues near the closure of a gap i e when the support of the equilibrium measure for large size random matrices has a gap that can be made to close the gaussian hermitian random matrix ensemble with external source described above has this feature the pearcey distribution is shown to satisfy a fourth order pde with cubic nonlinearity this also gives the pde for the transition probability of the pearcey process a limiting process associated with nonintersecting brownian motions on r c 2006 wiley periodicals inc

the analogues of entropy and of fisher s information measure in free probability theory i analogues of the entropy and fisher information measure for random variables in the context of free probability theory are introduced monotonicity properties and an analogue of the cramer rao inequality are proved

on existence and multiplicity of solutions of neumann boundary value problems for quasi linear elliptic equations abstract in the present paper the following neumann boundary value problem involving the p laplacian 1 1 p u f x u in u n 0 on is studied and some new existence and multiplicity results of solutions for the problem 1 1 are obtained

standing waves with a critical frequency for nonlinear schrodinger equations this paper is concerned with the existence and qualitative property of standing wave solutions for the nonlinear schrodinger equation with e being a critical frequency in the sense that we show that there exists a standing wave which is trapped in a neighbourhood of isolated minimum points of v and whose amplitude goes to 0 as moreover depending upon the local behaviour of the potential function v x near the minimum points the limiting profile of the standing wave solutions will be shown to exhibit quite different characteristic features this is in striking contrast with the non critical frequency case which has been extensively studied in recent years

existence and multiplicity results for quasilinear elliptic exterior problems with nonlinear boundary conditions abstract in this paper we study a class of quasilinear elliptic exterior problems with nonlinear boundary conditions existence of ground states and multiplicity results are obtained via variational methods

fluctuations at the edges of the spectrum of the full rank deformed gue we consider a full rank deformation of the gue frac w n sqrt n a n where a n is a full rank hermitian matrix of size n and w n is a gue the empirical eigenvalue distribution mu a n of a n converges to a probability distribution nu we identify all the possible limiting eigenvalue statistics at the edges of the spectrum including outliers edges and merging points of connected components of the limiting spectrum the results are stated in terms of a deterministic equivalent of the empirical eigenvalue distribution of w n a n namely the free convolution of the semi circle distribution and the empirical eigenvalues distribution of a n

anisotropic local laws for random matrices we develop a new method for deriving local laws for a large class of random matrices it is applicable to many matrix models built from sums and products of deterministic or independent random matrices in particular it may be used to obtain local laws for matrix ensembles that are anisotropic in the sense that their resolvents are well approximated by deterministic matrices that are not multiples of the identity for definiteness we present the method for sample covariance matrices of the form open image in new window where t is deterministic and x is random with independent entries we prove that with high probability the resolvent of q is close to a deterministic matrix with an optimal error bound and down to optimal spectral scales as an application we prove the edge universality of q by establishing the tracy widom airy statistics of the eigenvalues of q near the soft edges this result applies in the single cut and multi cut cases further applications include the distribution of the eigenvectors and an analysis of the outliers and bbp type phase transitions in finite rank deformations they will appear elsewhere we also apply our method to wigner matrices whose entries have arbitrary expectation i e we consider w a where w is a wigner matrix and a a hermitian deterministic matrix we prove the anisotropic local law for w a and use it to establish edge universality

universality of the local spacing distribution in certain ensembles of hermitian wigner matrices consider an n n hermitian random matrix with independent entries not necessarily gaussian a so called wigner matrix it has been conjectured that the local spacing distribution i e the distribution of the distance between nearest neighbour eigenvalues in some part of the spectrum is in the limit as n the same as that of hermitian random matrices from gue we prove this conjecture for a certain subclass of hermitian wigner matrices