On singular nonhomogeneous elliptic equations involving critical Caffarelli–Kohn–Nirenberg exponent

In this paper, we establish the existence of multiple solutions for nonhogeneous singular elliptic equations involving critical Caffarelli–Kohn–Nirenberg exponent, by using Ekeland’s Variational Principle and Mountain Pass Theorem without Palais Smale conditions.     

Solutions to nonlinear Schr?dinger equations with singular electromagnetic potential and critical exponent

We investigate existence and qualitative behavior of solutions to nonlinear Schr?dinger equations with critical exponent and singular electromagnetic potentials. We are concerned with magnetic vector potentials which are homogeneous of degree –1, including the Aharonov–Bohm class. In particular, by variational arguments we prove a result of multiplicity of solutions distinguished by symmetry properties.     

The Existence of Solutions of Elliptic Equations with Neumann Boundary Condition for Superlinear Problems

In this paper, we study and discuss the existence of multiple solutions of a class of non-linear elliptic equations with Neumann boundary condition, and obtain at least seven non-trivial solutions in which two are positive, two are negative and three are sign-changing. The study of problem (1.1):{-△u αu=f(u),x∈Ω, x∈Ω,δu/δr=0,x∈δΩ,is based on the variational methods and critical point theory. We form our conclusion by using the sub-sup solution method, Mountain Pass Theorem in order intervals, Leray-Schauder degree theory and the invariance of decreasing flow.     

Positive solutions for Neumann elliptic problems involving critical Hardy–Sobolev exponent with boundary singularities

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.     

Solutions with moving singularities for a semilinear parabolic equation

We consider the Cauchy problem for a semilinear heat equation with power nonlinearity. It is known that the equation has a singular steady state in some parameter range. Our concern is a solution with a moving singularity that is obtained by perturbing the singular steady state. By formal expansion, it turns out that the remainder term must satisfy a certain parabolic equation with inverse-square potential. From the well-posedness of this equation, we see that there appears a critical exponent. Paying attention to this exponent, for a prescribed motion of the singular point and suitable initial data, we establish the time-local existence, uniqueness and comparison principle for such singular solutions. We also consider solutions with multiple singularities.     

Blow-up for Strauss type wave equation with damping and potential

We study a kind of nonlinear wave equations with damping and potential, whose coefficients are both critical in the sense of the scaling and depend only on the spatial variables. Based on the earlier works, one may think there are two kinds of blow-up phenomenons when the exponent of the nonlinear term is small. It also means there are two kinds of law to determine the critical exponent. In this paper, we obtain a blow-up result and get the estimate of the upper bound of the lifespan in critical and sub-critical cases. All of the results support such a conjecture, although for now, the existence part is still open.     

具有临界增长的拟线性椭圆型混合边值问题的非平凡解

本文给出了ＲＮ中一类有界区域εαＮ上临界增长拟线性椭圆型方程Ｆｉ（ｘ，ｕ，Ｄｕ）－Ｆｕ（ｘ，ｕ，Ｄｕ）＝０的混合边值问题非平凡解的存在性。     

一类奇异临界双调和椭圆方程的群不变解

讨论一类含有Hardy-Sobolev临界指数项的奇异双调和椭圆方程,应用Lions集中紧性原理、Palais对称临界原理、Hardy-Rellich型不等式和变分方法,证明了方程在适当条件下群不变解的存在性和多重性.     

Existence result for a class of singular quasilinear elliptic equations with critical growth

In this paper, we obtain the existence of a nontrivial solution for a class of singular quasilinear elliptic equations with critical exponents. The proofs rely on a non-smooth critical point theory, and some techniques used by Brezis and Nirenberg.     

Estimates for the Sobolev trace constant with critical exponent and applications

In this paper we find estimates for the optimal constant in the critical Sobolev trace inequality that are independent of Ω. This estimates generalized those of Adimurthi and Yadava (Comm Partial Diff Equ 16(11):1733–1760, 1991) for general p. Here p _* : =  p(N  −  1)/(N  −  p) is the critical exponent for the immersion and N is the space dimension. Then we apply our results first to prove existence of positive solutions to a nonlinear elliptic problem with a nonlinear boundary condition with critical growth on the boundary, generalizing the results of Fernández Bonder and Rossi (Bull Lond Math Soc 37:119–125, 2005). Finally, we study an optimal design problem with critical exponent.      

半线性奇系数临界双调和方程的Dirichlet问题

主要探讨了两类半线性双调和Dirichlet问题：奇系数次临界问题和临界但带较弱奇性问题 ，得出了在临界维数和正常维数不同情况下都至少有一个正解的结论. 同时也研究了临界维数的消失问题，比较了奇系数与较弱奇性不同情况下临界维数的变化，得出奇性越大临界维数越少的结论.      

Renormalization and homogenization of fractional diffusion equations with random data

 This paper presents a renormalization and homogenization theory for fractional-in-space or in-time diffusion equations with singular random initial conditions. The spectral representations for the solutions of these equations are provided. Gaussian and non-Gaussian limiting distributions of the renormalized solutions of these equations are then described in terms of multiple stochastic integral representations. Received: 30 May 2000 / Revised version: 9 November 2001 / Published online: 10 September 2002 Mathematics Subject Classification (2000): Primary 62M40, 62M15; Secondary 60H05, 60G60 Key words or phrases: Fractional diffusion equation – Scaling laws – Renormalised solution – Long-range dependence – Non-Gaussian scenario – Mittag-Leffler function – Stable distributions – Bessel potential – Riesz potential     

Existence and Asymptotic Behavior of Ground State Solutions for Quasilinear Schr?dinger Equations with Unbounded Potential

The authors study the existence of standing wave solutions for the quasilinear Schr?dinger equation with the critical exponent and singular coefficients. By applying the mountain pass theorem and the concentration compactness principle, they get a ground state solution. Moreover, the asymptotic behavior of the ground state solution is also obtained.     

Elliptic problems with critical exponents and Hardy potentials

This paper is devoted to the existence of positive solutions of a Dirichlet problem with critical exponent and a singular potential. Under various assumption on the domain Ω, which include some kinds of unbounded domains, we prove the existence of ground states and of symmetric solutions.     

Life span and secondary critical exponent for degenerate and singular parabolic equations

In this paper, we investigate the large time behavior to the Cauchy problem of degenerate and singular parabolic equations. Firstly, we establish the secondary critical exponent on the decay asymptotic behavior of an initial value at infinity. Secondly, we give the large time behavior of the global solution. Finally, the precise estimate of life span for the blow-up solution is obtained.     

Carleman problem for two pairs of functions in a ring

We consider a particular case of the matrix Carleman problem for two pairs of functions in a ring and find a constructive solution of this problem. In addition, we propose an algorithm for the construction of solutions for two infinite systems of smooth transition and for a system of two singular equations of special type. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 5, pp. 662–671, May, 1997     

包含Sobolev-Hardy指数的p-Laplace方程的多解

在本文中，我们利用Sobolev-Hardy不等式，局部PS条件和亏格理论，证明了一类带临界Sobolev-Hardy指数的奇异p-Laplace方程存在多解．     

Positive solutions for Schrödinger–Poisson equations with a critical exponent

In this paper we establish the existence of a positive solution of the Schrödinger–Poisson equations with a critical Sobolev exponent. The methods used here are based on the concentration–compactness principle of P. L. Lions and methods of Brezis and Nirenberg.     

Simultaneous desingularizations of Calabi–Yau and special Lagrangian 3-folds with conical singularities. II. Examples

This article is a sequel to Chan (Ann Glob Anal Geom, to appear) on simultaneous desingularizations of Calabi–Yau and special Lagrangian (SL) 3-folds with conical singularities. In Chan (Ann Glob Anal Geom, to appear) we treated the question of starting with a conically singular Calabi–Yau 3-fold and an SL 3-fold with conical singularities at the same points and deforming both together to get a smooth situation. In this article, we survey the major result from Chan (Ann Glob Anal Geom, to appear) and describe some examples from our earlier articles (Chan Q J Math 57:151–181, 2006, Q J Math, to appear) on Calabi–Yau desingularizations. We then provide many explicit examples of Asymptotically Conical (AC) SL submanifolds in two specific AC Calabi–Yau manifolds. Using the result in Chan (Ann Glob Anal Geom, to appear), we construct smooth examples of compact SL 3-folds in compact Calabi–Yau 3-folds by gluing those AC SL 3-folds into some conically singular SL 3-folds at the singular points.     

On the efficiency of the WKB–Galerkin method in differential equations with variable coefficients

We have proposed an algorithm for the solution of inhomogeneous singular second-order differential equations with variable coefficients, based on a model of the hybrid WKB–Galerkin method. The efficiency of this approach is illustrated in the solution of an applied problem describing heat removal through a radiator of variable geometry. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 51, No. 1, pp. 82–87, January–March, 2008.