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In this paper, we discuss the validity of the result of Khuangsatung and Kangtunyakarn [Existence and convergence theorem for fixed point problem of various nonlinear mappings and variational inequality problems without some assumptions, Filomat 32(1) (2018) 305–309].

In a joint paper [4] by Otsubo, Terasoma and the first author, we proved that the special value 3F2(a,b,q;a+b,q;1) of the generalized hypergeometric function is a linear combination of log of algebraic numbers if the triplet (a,b,q) of rational numbers satisfies a certain numerical condition. However there remains a question how to obtain explicit descriptions of the values. In this paper, we give a method to do this, which is a further development of the technique in [4]., Comment: 22pages, To appear in Communications in Contemporary Mathematics

We demonstrate the asymptotic analysis of a semi-linear optimal control problem posed on a smooth oscillating boundary domain in the present paper. We have considered a more general oscillating domain than the usual “pillar-type” domains. Consideration of such general domains will be useful in more realistic applications like circular domain with rugose boundary. We study the asymptotic behavior of the problem under consideration using a new generalized periodic unfolding operator. Further, we are studying the homogenization of a non-linear optimal control problem and such non-linear problems are limited in the literature despite the fact that they have enormous real-life applications. Among several other technical difficulties, the absence of a sufficient criteria for the optimal control is one of the most attention-grabbing issues in the current setting. We also obtain corrector results in this paper.

In this paper, we prove Adams inequality with exact growth condition in the four-dimensional hyperbolic space ℍ4, ∫ℍ4 e32π2u2 − 1 (1 + |u|)2 dvg ≤ C∥u∥L2(ℍ4)2, (0.1) for all u ∈ Cc∞(ℍ4) with ∫ ℍ4(P2u)udvg ≤ 1. We will also establish an Adachi–Tanaka-type inequality in this setting. Another aspect of this paper is the Lions lemma in the hyperbolic space. We prove Lions lemma for the Moser functional and for a few cases of the Adams functional on the whole hyperbolic space.

In this paper, a modified iteration method (MIM) has been proposed to solve nonlinear second-order ODEs. Convergence analysis and error estimate of the proposed method are also discussed. Computational efficiency of this method is illustrated through numerical examples.

In this paper, we present new cubically convergent Newton-type iterative methods with dynamics for solving nonlinear algebraic equations under weak conditions. The proposed methods are free from second-order derivative and work well when [Formula: see text]. Numerical results show that the proposed method performs better when Newton’s method fails or diverges and competes well with same order existing method. Fractal patterns of different methods also support the numerical results and explain the compactness regarding the convergence, divergence, and stability of the methods to different roots.

We construct Ionel–Parker’s proposed refinement of the standard relative Gromov–Witten invariants in terms of abelian covers of the symplectic divisor and discuss in what sense it gives rise to invariants. We use it to obtain some vanishing results for the standard relative Gromov–Witten invariants. In a separate paper, we describe to what extent this refinement sharpens the usual symplectic sum formula and give further qualitative applications.

This paper is devoted to initial value problems for impulsive fractional differential equations of Caputo–Fabrizio type fractional derivative. By means of Banach’s fixed point theorem and Schaefer’s fixed point theorem, the existence and uniqueness results are obtained. Finally, an example is given to illustrate one of the main results.

In this paper, we introduce a new mixed type iterative process, which approximates the common fixed points of single-valued nonexpansive mappings and two multi-valued nonexpansive mappings in a uniformly convex Banach space. We establish strong and weak convergence theorems for the new iterative process in Banach space and give their corresponding applications.

Based on the approximations of the Lorenz system of differential equations from the papers [B. Zlatanovska and D. Dimovski, Systems of difference equations approximating the Lorentz system of differential equations, Contributions Sec. Math. Tech. Sci. Manu. XXXIII 1–2 (2012) 75–96, B. Zlatanovska and D. Dimovski, Systems of difference equations as a model for the Lorentz system, in Proc. 5th Int. Scientific Conf. FMNS, Vol. I (Blagoevgrad, Bulgaria, 2013), pp. 102–107, B. Zlatanovska, Approximation for the solutions of Lorenz system with systems of differential equations, Bull. Math. 41(1) (2017) 51–61], we define a Modified Lorenz system, that is a local approximation of the Lorenz system. It is a system of three differential equations, the first two are the same as the first two of the Lorenz system, and the third one is a homogeneous linear differential equation of fifth order with constant coefficients. The solution of this system is based on the results from [D. Dimitrovski and M. Mijatovic, A New Approach to the Theory of Ordinary Differential Equations (Numerus, Skopje, 1995), pp. 23–33].

The intention of this paper is to prove various stability results of reciprocal-septic and reciprocal-octic functional equations in non-Archimedean fields and nonzero real numbers relevant to Hyers, Rassias, and Găvruţa stability. Appropriate counter-examples are supplied to invalidate the results in the cases of singularities.

In this paper, a new approach and methodology is developed by incorporating differential quadrature technique with Bernstein polynomials. In differential quadrature method, approximations are done in a way that the derivatives of the function are replaced by a linear sum of functional values at the grid points of the given domain. In Bernstein differential quadrature method (BDQM), Bernstein polynomials are employed for spatial discretization so that a system of ordinary differential equations (ODE’s) is obtained which is solved by SSPRK-43 method. The stability of the method is also studied. The accuracy of the present method is checked by performing numerical experiments on two-dimensional coupled Burgers’ and Brusselator systems and fourth-order extended Fisher Kolmogorov (EFK) equation. Implementation of the method is very easy, efficient and capable of reducing the size of computational efforts.

In this paper, we study the existence and concentration behaviors of positive solutions to the following fractional nonlinear Schrödinger equation: [Formula: see text] where [Formula: see text] is a positive parameter, [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] is the fractional Laplacian. When the potential [Formula: see text] decays to zero like [Formula: see text], [Formula: see text], and [Formula: see text] like [Formula: see text] with [Formula: see text], we will show that the existence of ground states [Formula: see text] belonging to [Formula: see text], which concentrates at a minimum point of the auxiliary function [Formula: see text].

In this paper, we study the local well-posedness for the Cauchy problem of a semilinear fractional diffusion equation where the perturbations behave like [Formula: see text] and [Formula: see text], and [Formula: see text] is the characteristic function of a ball [Formula: see text]. Here, we are interested in the solvability of the problem when singular initial data [Formula: see text] are taken in [Formula: see text]. Eventually, we give sufficient conditions to the nonexistence of positive global solutions.

Everyone knows about the complicated solution of the nonlinear Fredholm integro-differential equation in general. Hence, often, authors attempt to obtain the approximate solution. In this paper, a numerical method for the solutions of the nonlinear Fredholm integro-differential equation (NFIDE) of the second kind in the complex plane is presented. In fact, by using the properties of Rationalized Haar (RH) wavelet, we try to give the solution of the problem. So far, as we know, no study has yet been attempted for solving the NFIDE in the complex plane. For this purpose, we introduce the continuous integral operator and real valued function. The Banach fixed point theorem guarantees that, under certain assumptions, the integral operator has a unique solution. Furthermore, we give an upper bound for the error analysis. An algorithm is presented to compute and illustrate the solutions for some numerical examples.

In this paper, we sharpen the lower bound on the codimension of the irreducible components of the Noether–Lefschetz locus of surfaces in projective toric threefolds given in [U. Bruzzo and A. Grassi, The Noether–Lefschetz locus of surfaces in toric threefolds, Commun. Contemp. Math. 20(5) (2018) 1–22]. We also provide a simpler proof of Theorem 4.11 in [U. Bruzzo and A. Grassi, The Noether–Lefschetz locus of surfaces in toric threefolds, Commun. Contemp. Math. 20(5) (2018) 1–22], which allows one to avoid some technical assumptions.

International audience; The subject matter of this paper concerns the derivation of fluid limits for gyro-kinetic models. The arguments apply for any collision kernel satisfying the usual conservations (mass, momentum, kinetic energy) and possessing a production entropy sign. We describe the set of equilibria in terms of several moments, we determine the average collision invariants, and we write the associated macro-scopic equations and the entropy inequality.

Let [Formula: see text] be a connected finite graph and [Formula: see text] be the set of functions defined on [Formula: see text]. Let [Formula: see text] be the discrete [Formula: see text]-Laplacian on [Formula: see text] with [Formula: see text] and [Formula: see text], where [Formula: see text] is positive everywhere. Consider the operator [Formula: see text]. We prove that [Formula: see text] is one-to-one, onto and preserves order. So it implies that there exists a unique solution to the equation [Formula: see text] for any given [Formula: see text]. We also prove that the equation [Formula: see text] has a solution which is unique up to a constant, where [Formula: see text] is the average of [Formula: see text]. With the help of these results, we finally give various conditions such that the [Formula: see text]th Kazdan–Warner equation [Formula: see text] has a solution on [Formula: see text] for given [Formula: see text] and [Formula: see text]. Thus we generalize Grigor’yan, Lin and Yang’s work Kazdan–Warner equation on graph, Calc. Var. Partial Differential Equations 55(4) (2016) Paper No. 92, 13 pp. for [Formula: see text] to any [Formula: see text].

In this paper, we propose a generalized [Formula: see text]-Laplacian Liénard type differential equation with a variable delay. By applying the Mawhin continuation theorem, we established a set of sufficient conditions on the existence of at least one periodic solution with period [Formula: see text]. It is significant that the growth degree with respect to the variables [Formula: see text] imposed on [Formula: see text] is allowed to be greater than [Formula: see text] and the growth degree with respect to the variables [Formula: see text] imposed on [Formula: see text] is allowed to be greater than [Formula: see text] so the result not only improves but also generalizes. Some examples are provided to illustrate the results.

This paper studies the global bifurcation phenomenon for the following homogeneous nonlocal eigenvalue problem [Formula: see text] Under some natural hypotheses on [Formula: see text] and [Formula: see text], we show that [Formula: see text] is a bifurcation point of the nontrivial solution set of the above problem. As application of the above result, we determine the interval of [Formula: see text], in which there exist positive solutions for the following Kirchhoff type problem [Formula: see text] where [Formula: see text] is asymptotically 3-linear at zero and infinity. Our results provide a positive answer to an open problem. Moreover, we also study the spectral structure for a homogeneous nonlocal eigenvalue problem.

This paper develops the notion and properties of the generalized principal eigenvalue for an elliptic system coupling an equation in a plane with one on a line in this plane, together with boundary conditions that express exchanges taking place between the plane and the line. This study is motivated by the reaction–diffusion model introduced by Berestycki, Roquejoffre and Rossi [The influence of a line with fast diffusion on Fisher–KPP propagation, J. Math. Biol. 66(4–5) (2013) 743–766] to describe the effect on biological invasions of networks with fast diffusion imbedded in a field. Here we study the eigenvalue associated with heterogeneous generalizations of this model. In a forthcoming work [Influence of a line with fast diffusion on an ecological niche, preprint (2018)] we show that persistence or extinction of the associated nonlinear evolution equation is fully accounted for by this generalized eigenvalue. A key element in the proofs is a new Harnack inequality that we establish for these systems and which is of independent interest.

To tackle a linear Fredholm integral equation on great interval, two numerical processes are involved: discretization and iterative scheme. The conventional numerical process is discretize first then use an iterative scheme as Jacobi’s method to approach the solutions of the huge algebraic system. In this paper, we propose an alternative numerical process, we apply an iterative scheme based on construction of a generalization of the iterative scheme for Jacobi method which is adapted to the system of linear bounded operators, then we use Nyström method to discretize only the diagonal part of the system. The convergence analysis of this new method is proved and numerical tests developed show its effectiveness.

The aim of this paper is to study the existence and uniqueness of solutions for a boundary value problem associated with a fractional nonlinear differential equation with higher order posed on the half-line. An appropriate continuous embedding for suitable Banach spaces are proved and the Minty–Browder theorem for monotone operators is used in the proof of existence of solutions for a boundary value problem of fractional order posed on the half-line.

In this paper, we consider an optimization model for solving the nonmonotone affine variational inequalities problem (AVI). It is formulated as a DC (Difference of Convex functions) program for which DCA (DC Algorithms) are applied. The resulting DCA are simple: it consists of solving successive convex quadratic program. Numerical experiments on several test problems illustrate the efficiency of the proposed approach in terms of the quality of the obtained solutions and the speed of convergence.

In this paper, we investigate spreading properties of the solutions of the Kolmogorov-Petrovsky-Piskunov-type, (to be simple,KPP-type) lattice system \begin{equation}\label{firstequation}\overset{.}u_{i}(t) =d^{\prime}_{i}(u_{i+1}(t)-u_{i}(t))+d_{i}(u_{i-1}(t)-u_{i}(t))+f(i,u_{i}).\end{equation} we develop some new discrete Harnack-type estimates and homogenization techniques for this lattice system to construct two speeds $\overline\omega \leq \underline \omega$ such that $\displaystyle{\lim_{t\rightarrow+\infty}}\sup \limits_{i\geq\omega t}|u_i(t)|=0$ for any $\omega>\overline{\omega}$, and $\displaystyle{\lim_{t\rightarrow+\infty}}\sup \limits_{0\leq i\leq\omega t}|u_i(t)-1|=0$ for any $\omega

In this paper, we propose a combinatorial approach to study Calabi–Yau threefolds constructed as a resolution of singularities of a double covering of [Formula: see text] branched along an arrangement of eight planes. We use this description to give a complete classification of arrangements of eight planes in [Formula: see text] defining Calabi–Yau threefolds modulo projective transformation with [Formula: see text] and to derive their geometric properties (Kummer surface fibrations, automorphisms, special elements in families).

In this paper, we use the variational iteration method (VIM) to construct approximate solutions for the general [Formula: see text]th-order integro-differential equations. We show that his method can be effectively and easily used to solve some classes of linear and nonlinear Volterra integro-differential equations. Finally, some numerical examples with exact solutions are given.

In this paper, we study some type of equations which may model the behavior of species inhabiting in some habitat. For our purpose, using a priori bounded techniques, we obtain a positive solution to a family of non-local partial differential equations with non-homogeneous boundary conditions.

This paper presents a numerical method for solving a certain class of Fredholm integral equations of the first kind, whose unknown function is singular at the end-points of the integration domain, and has a weakly singular logarithmic kernel with analytical treatments of the singularity. To achieve this goal, the kernel is parametrized, and the unknown function is assumed to be in the form of a product of two functions; the first is a badly-behaved known function, while the other is a regular unknown function. These two functions are approximated by using the economized monic Chebyshev polynomials of the same degree, while the given potential function is approximated by monic Chebyshev polynomials of the same degree. Further, the two parametric functions associated to the parametrized kernel are expanded into Taylor polynomials of the first degree about the singular parameter, and an asymptotic expression is created, so that the obtained improper integrals of the integral operator become convergent integrals. Thus, and after using a set of collocation points, the required numerical solution is found to be equivalent to the solution of a linear system of algebraic equations. From the illustrated example, it turns out that the proposed method minimizes the computational time and gives a high order accuracy.

In this paper, the minimization problem [Formula: see text] where [Formula: see text] is studied when [Formula: see text] ([Formula: see text]) is an open, bounded, convex domain with smooth boundary and [Formula: see text]. We show that [Formula: see text] is either zero, when the maximum of the distance function to the boundary of [Formula: see text] is greater than [Formula: see text], or it is a positive real number, when the maximum of the distance function to the boundary of [Formula: see text] belongs to the interval [Formula: see text]. In the latter case, we provide estimates for [Formula: see text] and show that for [Formula: see text] sufficiently large [Formula: see text] coincides with the principal frequency of the [Formula: see text]-Laplacian in [Formula: see text]. Some particular cases and related problems are also discussed.

This paper not only presents a criterion for the Cordes–Nirenberg space [Formula: see text] imbedding between the associate Morrey space [Formula: see text] and the Morrey space [Formula: see text], but also characterizes a Radon measure [Formula: see text] such that the Cordes–Nirenberg potential space [Formula: see text] is restricted to the [Formula: see text]-based Campanato space [Formula: see text] — thereby giving an application of the discovered characterization to the regularity for a class of the elliptic equations with symmetric [Formula: see text]-coefficients.

Dispersive SQG equation have been studied by many works (see, e.g., [M. Cannone, C. Miao and L. Xue, Global regularity for the supercritical dissipative quasi-geostrophic equation with large dispersive forcing, Proc. Londen. Math. Soc. 106 (2013) 650–674; T. M. Elgindi and K. Widmayer, Sharp decay estimates for an anisotropic linear semigroup and applications to the surface quasi-geostrophic and inviscid Boussinesq systems, SIAM J. Math. Anal. 47 (2015) 4672–4684; A. Kiselev and F. Nazarov, Global regularity for the critical dispersive dissipative surface quasi-geostrophic equation, Nonlinearity 23 (2010) 549–554; R. Wan and J. Chen, Global well-posedness for the 2D dispersive SQG equation and inviscid Boussinesq equations, Z. Angew. Math. Phys. 67 (2016) 104]), which is very similar to the 3D rotating Euler or Navier–Stokes equations. Long time stability for the dispersive SQG equation without dissipation was obtained by Elgindi–Widmayer [Sharp decay estimates for an anisotropic linear semigroup and applications to the surface quasi-geostrophic and inviscid Boussinesq systems, SIAM J. Math. Anal. 47 (2015) 4672–4684], where the initial condition [Formula: see text] [Formula: see text] plays a important role in their proof. In this paper, by using the Strichartz estimate, we can remove this initial condition. Namely, we only assume the initial data is in the Sobolev space like [Formula: see text]. As an application, we can also obtain similar result for the 2D Boussinesq equations with the initial data near a nontrivial equilibrium.

In this paper, existence and multiplicity of positive solutions of the elliptic system − Δu + V1(x)u = μ1(x)u3 + β(x)uv2in Ω 𝜀, − Δv + V2(x)v = β(x)u2v + μ 2(x)v3 in Ω 𝜀,u = v = 0 on ∂Ω𝜀 is proved, where Ω𝜀 is an exterior domain in ℝN such that ℝN∖Ω 𝜀 is far away from the origin and contains a sufficiently large ball, N = 1, 2, 3, and the coefficients Vj,μj,β are continuous functions on ℝN which tend to positive constants at infinity. We do not assume μ1,μ2,β to be positive functions.

This paper is devoted to study some chaotic properties of iterated function systems (IFSs). Specially, a new notion named thick chaotic IFSs is introduced. The relationship between thick chaos and another properties of some notions in dynamical systems are studied.

In this paper, Pachpatte’s inequality is employed to discuss the Ulam–Hyers stabilities for Volterra integrodifferential equations and Volterra delay integrodifferential equations in Banach spaces on both finite and infinite intervals. Examples are given to show the applicability of our obtained results.

This paper concerns a global controllability problem for heat equations. In the absence of control, the solution to the linear heat system globally exists. While for each initial data, we can find a feedback control acting on an internal subset of the space domain such that the corresponding solution to the system blows up at given time.

In this paper, we derive the following Leray–Trudinger type inequality on a bounded domain [Formula: see text] in [Formula: see text] containing the origin. [Formula: see text] [Formula: see text] Here, [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], [Formula: see text] for [Formula: see text] This improves an earlier result by Psaradakis and Spector. Also, we prove that for any [Formula: see text] in the place of [Formula: see text], the above inequality is false if we take [Formula: see text].

In this paper, we study an evolution equation involving the normalized [Formula: see text]-Laplacian and a bounded continuous source term. The normalized [Formula: see text]-Laplacian is in non-divergence form and arises for example from stochastic tug-of-war games with noise. We prove local [Formula: see text] regularity for the spatial gradient of the viscosity solutions. The proof is based on an improvement of flatness and proceeds by iteration.

In this paper, we are concerned with the following nonlinear Choquard equation [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text]. If [Formula: see text] lies in a gap of the spectrum of [Formula: see text] and [Formula: see text] is of critical growth due to the Hardy–Littlewood–Sobolev inequality, we obtain the existence of nontrivial solutions by variational methods. The main result here extends and complements the earlier theorems obtained in [N. Ackermann, On a periodic Schrödinger equation with nonlocal superlinear part, Math. Z. 248 (2004) 423–443; B. Buffoni, L. Jeanjean and C. A. Stuart, Existence of a nontrivial solution to a strongly indefinite semilinear equation, Proc. Amer. Math. Soc. 119 (1993) 179–186; V. Moroz and J. Van Schaftingen, Existence of groundstates for a class of nonlinear Choquard equations, Trans. Amer. Math. Soc. 367 (2015) 6557–6579].

In this paper, we characterize a family of solitary waves for nonlinear Schrödinger equation (NLS) with derivative (DNLS) by the structure analysis and the variational argument. Since DNLS does not enjoy the Galilean invariance any more, the structure analysis here is closely related with the nontrivial momentum and shows the equivalence of nontrivial solutions between the quasilinear and the semilinear equations. Firstly, for the subcritical parameters [Formula: see text] and the critical parameters [Formula: see text], we show the existence and uniqueness of the solitary waves for DNLS, up to the phase rotation and spatial translation symmetries. Secondly, for the critical parameters [Formula: see text], [Formula: see text] and the supercritical parameters [Formula: see text], there is no nontrivial solitary wave for DNLS. At last, we make use of the invariant sets, which is related to the variational characterization of the solitary wave, to obtain the global existence of solution for DNLS with initial data in the invariant set [Formula: see text], with [Formula: see text], [Formula: see text] or [Formula: see text]. On the one hand, different with the scattering result for the [Formula: see text]-critical NLS in [B. Dodson, Global well-posedness and scattering for the mass critical nonlinear Schrödinger equation with mass below the mass of the ground state, Adv. Math. 285(5) (2015) 1589–1618], the scattering result of DNLS does not hold for initial data in [Formula: see text] because of the existence of infinity many small solitary/traveling waves in [Formula: see text] with [Formula: see text], [Formula: see text] or [Formula: see text]. On the other hand, our global result improves the global result in [Y. Wu, Global well-posedness of the derivative nonlinear Schrödinger equations in energy space, Anal. Partial Differential Equations 6(8) (2013) 1989–2002; Global well-posedness on the derivative nonlinear Schrödinger equation, Anal. Partial Differential Equations 8(5) (2015) 1101–1112] (see Corollary 1.6).

In this paper, by using a modified Nehari–Pankov manifold, we prove the existence and the asymptotic behavior of least energy solutions for the following indefinite biharmonic equation: [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text] is a parameter, [Formula: see text] is a nonnegative potential function with nonempty zero set [Formula: see text], [Formula: see text] is a positive function such that the operator [Formula: see text] is indefinite and non-degenerate for [Formula: see text] large. We show that both in subcritical and critical cases, equation [Formula: see text] admits a least energy solution which for [Formula: see text] large localized near the zero set [Formula: see text].

Let [Formula: see text] and [Formula: see text] be the local Hardy space in the sense of D. Goldberg. In this paper, the authors establish two bilinear decompositions of the product spaces of [Formula: see text] and their dual spaces. More precisely, the authors prove that [Formula: see text] and, for any [Formula: see text], [Formula: see text], where [Formula: see text] denotes the local BMO space, [Formula: see text], for any [Formula: see text] and [Formula: see text], the inhomogeneous Lipschitz space and [Formula: see text] a variant of the local Orlicz–Hardy space related to the Orlicz function [Formula: see text] for any [Formula: see text] which was introduced by Bonami and Feuto. As an application, the authors establish a div-curl lemma at the endpoint case.

In this paper, we study the existence of positive multi-peak solutions to the fractional Schrödinger–Poisson system [Formula: see text] where [Formula: see text] is a small parameter, [Formula: see text] is a positive function, [Formula: see text] and [Formula: see text] Under some given conditions which are given in Sec. ??, we prove the existence of a positive solution with m-peaks and concentrating near a given local maximum point of [Formula: see text]

In this paper, we give new conditions for existence and uniqueness of best proximity point. Also, we introduce the concept of cyclic contraction and nonexpansive for multivalued map and we give existence and convergence theorems for best proximity point in the complete metric space.

In this paper, we consider the following singular elliptic problem involving an exponential nonlinearity in two dimensions: [Formula: see text] [Formula: see text] where [Formula: see text] is a bounded domain with smooth boundary, [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. We show the existence and multiplicity of positive solutions globally with respect to the bifurcation parameter [Formula: see text].

In this paper, we study a [Formula: see text]th-order impulsive eigenvalue problem with an advanced argument. We shall establish several criteria for the optimal intervals of the parameter [Formula: see text] so as to ensure existence of single or many positive solutions. Our methods are based on transformation technique, Hölder’s inequality and the eigenvalue theory.

We consider a sequence of blowup solutions of a two dimensional, second order elliptic equation with exponential nonlinearity and singular data. This equation has a rich background in physics and geometry. In a work of Bartolucci-Chen-Lin-Tarantello it is proved that the profile of the solutions differs from global solutions of a Liouville type equation only by a uniformly bounded term. The present paper improves their result and establishes an expansion of the solutions near the blowup points with a sharp error estimate., Comment: 21 pages. Communications on contemporary mathematics, in press

In this article we study a class of mixed monotone operators with convexity on ordered Banach spaces and present some new tripled fixed point theorems by means of partial order theory, we get the existence and uniqueness of tripled fixed points without assuming the operator to be compact or continuous, which extend the existing corresponding results. As applications, we utilize the results obtained in this paper to study the existence and uniqueness of positive solutions for a fractional differential equation boundary value problem.

In this paper, we prove the global gradient estimates on the generalized Lebesgue spaces for weak solutions to elliptic quasilinear obstacle problems. It is worth noticing that the coefficients related to the obstacle problems are merely measurable with small BMO norms and the underlying domain does not satisfy any smoothness conditions.

In this paper, we prove the local existence and uniqueness of the solution with discontinuous density for the inhomogeneous non-resistive magnetohydrodynamics (MHD) equations on a [Formula: see text] bounded domain [Formula: see text] or [Formula: see text], if the initial data [Formula: see text] with [Formula: see text] satisfies [Formula: see text].