On a perturbed anisotropic equation with a critical exponent

Abstract We study a perturbed anisotropic equation without using the knowledge of the limiting problem. This provides a different method from that introduced by Brézis and Nirenberg [4]. Our arguments use some tools recently developed in [5, 6]. Keywords: Anisotropic critical exponent, Critical level, Compactness, Nehari manifold Mathematics Subject Classification (2000): 35J25, 35J60, 35J65     

On a periodic Schrödinger equation with nonlocal superlinear part

We consider the Choquard-Pekar equation and focus on the case of periodic potential V. For a large class of even functions W we show existence and multiplicity of solutions. Essentially the conditions are that 0 is not in the spectrum of the linear part –+V and that W does not change sign. Our results carry over to more general nonlinear terms in arbitrary space dimension N2.Mathematics Subject Classification (2000):35Q55, 35Q40, 35J10, 35J20, 35J60, 46N50, 49J35, 81V70in final form: 14 November 2003     

Existence and multiplicity of nontrivial solutions for an elliptic equation on ℝ<Superscript><Emphasis Type="Italic">N</Emphasis></Superscript> with indefinite linear part

 Via the Linking Theorem and Pseudo-index theory, we consider the existence and multiplicity of nontrivial solutions for a class of elliptic problems in all of ℝ ^N with indefinite linear part involving resonance and non-resonance at any eigenvalue. Received: 9 September 2002 / Revised version: 14 February 2003 Published online: 24 April 2003 Mathematics Subject Classification (2000): 35J20, 35J70     

On the convergence of genetic algorithms – a variational approach

A variational approach is introduced to study the existence and uniqueness of stationary states and (exponential) stability of genetic algorithms with mutation and interactive selection.Mathematics Subject Classification (2000): 35J20 (90C30, 92D25, 35J60, 31C25)     

The Paneitz Curvature Problem on Lower-Dimensional Spheres

In this paper we prescribe a fourth order conformal invariant (the Paneitz curvature) on the n-spheres, with n∊{5,6}. Using dynamical and topological methods involving the study of critical points at infinity of the associated variational problem, we prove some existence results. Mathematics Subject Classifications (2000): 35J60, 53C21, 58J05, 35J30.     

Bifurcation phenomena associated to a class of p-Laplacian like operators

 Global bifurcation of positive solutions for some degenerate quasilinear elliptic problems is considered. The uniform estimate of the gradient of weak solutions is given. This estimate is crucial in our arguments. Received: 10 August 2001 Supported in part by Grant-in-Aid for Scientific Research (No. 11640207), Ministry of Education, Science, Sports and Culture, Japan. Mathematics Subject Classification (2000): 35B32, 35J25, 35J70     

Green function for X-elliptic operators

We study the Green function associated to second order X-elliptic operators with non-regular coefficients. An existence and uniqueness theorem is given, along with some local estimations and a representation formula for the solution of the Dirichlet problem.Mathematics Subject Classification (2000): 34B27, 35H99, 35J70, 35C15Revised version: 28 June 2004     

Stability remarks to the obstacle problem for p-Laplacian type equations

We extend some convergence and L ¹ stability results for the coincidence set to the p-obstacle problem under natural nondegeneracy conditions and without restrictions on p, . We rely on the local regularity of the solution and, as an application, we show the existence of a solution to the thermal membrane problem, and in a limit nonlocal case also its uniqueness for small data.Received: 2 October 2003, Accepted: 10 May 2004, Published online: 10 December 2004Mathematics Subject Classification (2000): 35J85, 49J40, 35R35, 47J20     

On the existence of periodic solutions in the thermistor problem with degenerate thermal conductivity

Abstract This paper deals with the existence of weak periodic solutions for a parabolic-elliptic system proposed as a model for a time dependent thermistor with degenerate thermal conductivity. Applying the maximal monotone mappings theory, we prove an existence result for weak periodic solutions. Keywords: Nonlinear parabolic-elliptic system of degenerate type, Periodic solutions, Thermistor problem Mathematics Subject Classification (2000): 35B10, 35J60, 35K65     

Marino—Prodi perturbation type results and Morse indices of minimax critical points for a class of functionals in Banach spaces

In this work, we study a class of Euler functionals defined in Banach spaces, associated with quasilinear elliptic problems involving p-Laplace operator (p > 2). First we obtain perturbation results in the spirit of the remarkable paper by Marino and Prodi (Boll. U.M.I. (4) 11(Suppl. fasc. 3): 1–32, 1975), using the new definition of nondegeneracy given in (Ann. Inst. H. Poincaré: Analyse Non Linéaire. 2:271–292, 2003). We also extend Morse index estimates for minimax critical points, introduced by Lazer and Solimini (Nonlinear Anal. T.M.A. 12:761–775, 1988) in the Hilbert case, to our Banach setting. Mathematics Subject Classification (1991) 58E05, 35B20, 35J60, 35J70     

A general theory of structure spaces with applications to spaces of prime ideals

A structure space is a quadrupleX=(X, d, A, P), where for some setR, X A=2 ^R ,d:X×X A is defined byd(I, J)=J–I, andP is the family of cofinite subsets ofR. Forr P, I X, N _r (I)={J X: d(I, J) r},To(X)={Q X: if x Q there is anr P such thatN _r (x) Q}. ThenTo(X) is a (not usually Hausdorff) topology onX called the hull-kernel topology. Replacing d byd ^*, whered ^* (I, J)=d(J, I), or byd ^s, whered ^s (I, J.)=d(I, J) d ^* (I, J), and proceeding in the obvious way yields thedual hull-kernel topology To(X ^*) andsymmetric topology To(X ^s ). The latter is always a zero-dimensional Hausdorff space. When R is a commutative ring with identity andX is a collection of proper prime ideals ofR, To(X ^s ) is usually called thepatch topology. Our generality enables us to improve on known results in the case of space of prime ideals and to apply this theory to a wide variety of algebraic structures. In particular, we establish criteria for a subspace of a structure space to be closed in the symmetric topology; we establish a duality between families of maximal elements in the hull-kernel topology and families of minimal elements in the dual hull-kernel topology of subspaces that are closed in the symmetric topology; we use topological constructions to generalize certain ring theoretic notions, such as radical ideals an annihilator ideals; we use this theory to obtain new results about subspaces of the space prime ideals of a reduced, commutative ring.Presented by F. E. J. Linton.This author's research was supported by a grant from the CUNY-PSC research award program.     

On the noncommutative residue and the heat trace expansion on conic manifolds

 Given a cone pseudodifferential operator P we give a full asymptotic expansion as t → 0 ⁺ of the trace Tr Pe ^-tA , where A is an elliptic cone differential operator for which the resolvent exists on a suitable region of the complex plane. Our expansion contains log t and new (log t)² terms whose coefficients are given explicitly by means of residue traces. Cone operators are contained in some natural algebras of pseudodifferential operators on which unique trace functionals can be defined. As a consequence of our explicit heat trace expansion, we recover all these trace functionals. Received: 12 November 2001 / Revised version: 26 June 2002 Mathematics Subject Classification (2000): Primary 58J35; Secondary 35C20, 58J42     

Non-linear elliptic problems having natural growth and <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript> data in Orlicz spaces

An existence result for a strongly non-linear elliptic equation with natural growth condition on the non-linearity and L¹-data is proved. Mathematics Subject Classification (2000) 35J25, 35J60     

Sub-Markovian resolvents under weak duality hypothesis

 We study the weak duality between two sub-Markovian resolvents of kernels on a Lusin topological space with respect to a given measure m. Our frame covers the probabilistic context of two Borel Markov processes in weak duality. The main results are related to: the coincidence of the m-semipolar and m-cosemipolar sets, the Revuz correspondence between the measures charging no cofinely open m-polar set and the potential kernels associated with the homogeneous random measures of the process, the equivalence between smoothness and cosmoothness (a smooth measure is the Revuz measure of a continuous additive functional). We extend and improve results of R.M. Blumenthal-R.K. Getoor, J. Azéma, D. Revuz, J.B. Walsh, R.K. Getoor-M.J. Sharpe. Received: 24 July 2001 / Revised version: 29 December 2002 / Published online: 12 May 2003 Supported by the CNCSIS grant no. 33518/2002 (code 431), the CERES program of the Romanian Ministry of Ed. and Research, contract no. 152/2001 and the EURROMMAT program ICA1-CT-2000-70022 of the European Commission. Mathematics Subject Classification (2000): 60J40, 60J45, 60J55, 60J35, 31C15, 31C25, 31D05     

Semicontinuity of the solution mappings to weak generalized parametric Ky Fan inequality problems with trifunctions

In this article, we discuss the lower semicontinuity of solution maps without the condition of C-strict monotonicity for two classes of weak generalized parametric Ky Fan inequalities under the case that the f-solution set be a general set-valued one. Our results extend the recent ones in the literature (e.g. Cheng, Y.H., Zhu, D.L.: Global stability results for the weak vector variational inequality. J. Glob. Optim. 32, 543–550 (2005); C.R. Chen and S.J. Li, On the solution continuity of parametric generalized systems, Pac. J. Optim. 6 (2010), pp. 141–151; Gong, X.H., Yao, J.C.: Lower semicontinuity of the set of efficient solutions for generalized systems. J. Optim. Theory Appl. 138, 197–205 (2008); Gong, X.H.: Continuity of the solution set to parametric weak vector equilibrium problems. J. Optim. Theory Appl. 139, 35–46 (2008)). Several examples are given for the illustration of our results.     

Geometric Heat Comparison Criteria for Riemannian Manifolds

The main results of this paper are small-time heat comparison results for two points in two manifolds with characteristic functions as initial temperature distributions (Theorems 1 and 2). These results are based on the geometric concepts of (essential) distance from the complement and spherical area function. We also discuss some other geometric results about the heat development and illustrate them by examples. Mathematics subject classifications (2000): 58J35, 35K05     

Nonlinear evolutions with Carathéodory forcing

Let A be an m-accretive operator in a real Banach space X and f : J x X X a function of Carathéodory type, where $ J = [0, a] \subset \mathbb{R} $. This paper investigates the existence of mild solutions of the evolution system satisfying additional time-dependent constraints $ u(t) \in K(t) $ on J for a given tube K(·). Main emphasis is on existence results that are valid under minimal assumptions on f, K and X.AMS Subject Classification: Primary: 47J35, 35K90, Secondary: 34G25, 47H20.     

Existence and nonexistence results for quasilinear elliptic equations involving the <Emphasis Type="Italic">p</Emphasis>-Laplacian with a critical potential

This paper deals with the existence and nonexistence results for quasilinear elliptic equations of the form -_pu=f(x, u), where _p:=div(|u|^p-2u), p>1, and the solutions are understood in the sense of renormalized or, equivalently, entropy solutions. In particular we prove nonexistence results in the case f(x,u)=u^p|x|^-p, that is related to a classical Hardy inequality. Mathematics Subject Classification (2000) 35D05, 35D10, 35J20, 35J25, 35J70     

Existence of Conformal Metrics on Spheres with Prescribed Paneitz Curvature

In this paper we study the problem of prescribing a fourth order conformal invariant (the Paneitz curvature) on the n-sphere, with n 5. Using tools from the theory of critical points at infinity, we provide some topological conditions on the level sets of a given positive function under which we prove the existence of a metric, conformally equivalent to the standard metric, with prescribed Paneitz curvature.Mathematics Subject Classification (2000): 35J60, 53C21, 58J05Send offprint requests to: Khalil El Mehdi     

Realizations of Differential Operators on Conic Manifolds with Boundary

We study the closed extensions (realizations) of differential operators subject to homogeneous boundary conditions on weighted L _p -Sobolev spaces over a manifold with boundary and conical singularities. Under natural ellipticity conditions we determine the domains of the minimal and the maximal extension. We show that both are Fredholm operators and give a formula for the relative index. Mathematics Subject Classifications (2000): Primary 58J32; Secondary 35G70, 35S15.