Showing papers in "Complex Variables and Elliptic Equations in 2016"
TL;DR: In this article, the Mittag-Leffler functions with their normalization are considered and sufficient conditions are obtained so that they have certain geometric properties including univalency, starlikeness, convexity and close-to-convexity in the open unit disk.
Abstract: In the present investigation, the Mittag-Leffler functions with their normalization are considered. Several sufficient conditions are obtained so that the Mittag-Leffler functions have certain geometric properties including univalency, starlikeness, convexity and close-to-convexity in the open unit disk. Partial sums of Mittag-Leffler functions are also studied. The results obtained are new and their usefulness is depicted by deducing several interesting corollaries and examples.
59 citations
TL;DR: In this article, a representation of the Green's function of the classical Neumann problem for the Poisson equation in the unit ball of arbitrary dimension is given, which can be expressed in terms of elementary functions and its explicit form can be written out.
Abstract: Representation of the Green’s function of the classical Neumann problem for the Poisson equation in the unit ball of arbitrary dimension is given. In constructing this function, we use the representation of the fundamental solution of the Laplace equation in the form of a series. It is shown that the Green’s function can be represented in terms of elementary functions and its explicit form can be written out. An explicit form of the Neumann kernel at and .
41 citations
TL;DR: In this article, the authors studied the problem of finding at least two positives solutions for a bounded smooth domain, where the fibering maps are combined with the Nehari manifold to show the existence of such a problem.
Abstract: The purpose of this work is to study the following singular problem:where , be a bounded smooth domain, is a positive parameter, such that and where and r is the homogeneity degree of the function f. Under appropriate assumptions on the function f, we employ the method of Nehari manifold combined with the fibering maps in order to show the existence of such that for all , problem has at least two positives solutions.
41 citations
TL;DR: In this article, two classes of linear equations of discrete convolution type with harmonic singular operator were investigated using the Laurent transform theory, and the solutions of the equations were obtained in the class of Holder continuous functions.
Abstract: In this paper, we investigate two classes of linear equations of discrete convolution type with harmonic singular operator. Using the Laurent transform theory, we turn the above linear equations into Riemann boundary value problems. Then, the solutions of the equations are obtained in the class of Holder continuous functions.
33 citations
TL;DR: In this paper, the authors studied the class of univalent logharmonic mappings in the unit disk, where and are analytic in and is a normalized starlike analytic function.
Abstract: This paper studies the class consisting of univalent logharmonic mappings in the unit disk , where and are analytic in and is a normalized starlike analytic function. A representation theorem for these mappings is obtained, which yields sharp distortion estimates, and a sharp Bohr radius.
32 citations
TL;DR: In this article, Hardy-type inequalities for a large class of sub-elliptic operators that belong to the class of -Laplacians were derived and the constants for the constants involved were given.
Abstract: We derive Hardy-type inequalities for a large class of sub-elliptic operators that belong to the class of -Laplacians and find explicit values for the constants involved. Our results generalize previous inequalities obtained for Grushin-type operatorswhich were proved to be sharp.
32 citations
TL;DR: In this paper, the existence and multiplicity of positive radial solutions for two classes of nonlocal elliptic systems defined on bounded annular domains or exterior domains is studied. But the authors focus on the second-order functional ordinary elliptic system.
Abstract: Using fixed point techniques, we study the existence and multiplicity of positive radial solutions for two classes of nonlocal elliptic systems defined on bounded annular domains or exterior domains. To this end, we reduce our problem to second-order functional ordinary elliptic systems. Our approach also allows us to study systems involving various orders, which serve as models for the suspension bridge equations.
31 citations
TL;DR: In this article, the authors studied the existence of nonsmooth weak solutions for the boundary value problem, where the nonlinearity is subcritical growth and does not satisfy the Ambrosetti-Rabinowitz (AR) condition.
Abstract: We study the existence of nontrivial weak solutions for the following boundary value problemwhere is a bounded domain in , is a strongly degenerate elliptic operator, the nonlinearity is subcritical growth and does not satisfy the Ambrosetti–Rabinowitz (AR) condition.
30 citations
TL;DR: In this article, a list of open problems in pluripotential theory partially motivated by their applications to complex differential geometry is proposed, which includes both local questions as well as issues related to the compact complex manifold setting.
Abstract: We propose a list of open problems in pluripotential theory partially motivated by their applications to complex differential geometry. The list includes both local questions as well as issues related to the compact complex manifold setting.
28 citations
TL;DR: In this article, Liouville-type theorems for the elliptic system of inequalities were established for the case where the strongly degenerate operator of the formwhere satisfies some growth conditions.
Abstract: We establish Liouville-type theorems for the elliptic system of inequalitiesHere satisfy some growth conditions, and is the strongly degenerate operator of the formwhere satisfies some certain conditions. As a direct consequence, we also obtain the Liouville-type theorem for the following elliptic inequality
24 citations
TL;DR: In this paper, the existence of standing wave solutions for a class of quasilinear Schrodinger equations with subcritical or critical growth is studied. And the authors investigate the range of parameters for which non-trivial solutions can be guaranteed.
Abstract: We study the existence of standing wave solutions for a class of quasilinear Schrodinger equations with subcritical or critical growthwhere is a given potential, is a parameter. By variational methods, we investigate the range of parameters where the existence of non-trivial solutions can be guaranteed. Especially, by combing Resonance and Hahn–Banach theorems, we improve some previous results on .
TL;DR: In this paper, the existence and regularity of the solutions to the following anisotropic problem, involving a singular nonlinearity involving a bounded regular domain in and, were proved using an approximation approach.
Abstract: Using an approximation approach, we prove the existence and regularity of the solutions to the following anisotropic problem, involving a singular nonlinearitywhere is a bounded regular domain in and , we will assume without loss of generality that and that f is a non-negative function belonging to a suitable Lebesgue space
TL;DR: In this paper, a class of non-local equations involving the fractional p-Laplacian, where the non-linear term is assumed to have critical exponential growth, is considered.
Abstract: This paper deals with a class of non-local equations involving the fractional p-Laplacian, where the non-linear term is assumed to have critical exponential growth. More specifically, by applying variational methods together with a suitable Trudinger-Moser inequality for fractional Sobolev space, we obtain the existence of at least two positive weak solutions.
TL;DR: In this article, the authors studied properties of open, discrete ring mappings satisfying generalized modular inequalities, which generalize the known class of quasiregular mappings and their extensions known as mappings of finite distortion.
Abstract: We study properties of open, discrete ring mappings satisfying generalized modular inequalities. Such mappings generalize the known class of quasiregular mappings and their extensions known as mappings of finite distortion. We apply our results to open discrete ring mappings satisfying condition (N) and having local inverses, and we focus especially on the case . We show that such mappings cannot have essential singularities and also that Zoric’s theorem can hold in this case under some conditions even if . This is in contrast even with the known case of quasiregular mappings.
TL;DR: In this article, the existence of multiple solutions for impulsive fourth-order differential equations of Kirchhoff type was studied using a variational method and some critical points theorems.
Abstract: In this paper, we study the existence of multiple solutions for impulsive fourth-order differential equations of Kirchhoff type. Using a variational method and some critical points theorems, we obtain some new criteria for guaranteeing that impulsive fourth-order differential equations of Kirchhoff type have three and infinitely many solutions. Some recent results are extended and improved. Some examples are presented to demonstrate the applications of our main results.
TL;DR: In this paper, the authors introduced and studied the mixed norm variable order Bergman space on the unit disc in the complex plane and proved the boundedness of the Bergman projection and revealed the dependence of the nature of such spaces on possible growth of variable exponent p(r) when from inside the interval.
Abstract: We introduce and study the mixed norm variable order Bergman space on the unit disc in the complex plane. The mixed norm variable order Lebesgue-type space is defined by the requirement that the sequence of the variable exponent -norms of the Fourier coefficients of the function f belongs to Then is defined to be the subspace of which consists of analytic functions. We prove the boundedness of the Bergman projection and reveal the dependence of the nature of such spaces on possible growth of variable exponent p(r) when from inside the interval The situation is quite different in the cases and In the case we also characterize the introduced Bergman space as the space of Hadamards’s fractional derivatives of functions from the Hardy space The case is specially studied, and an open problem is formulated in this case. We also reveal the conditions on the rate of growth of p(r) when when isometrically, and when this is not longer true.
TL;DR: In this paper, the capacity and Cegrell's classes of m-subharmonic functions associated to any m-positive closed current T were introduced and investigated, and a generalization of the Demailly and Xing results about the definition and continuity of the complex hessian operator was presented.
Abstract: In this paper, we firstly introduce the concepts of capacity and Cegrell’s classes of m-subharmonic functions , , associated to any m-positive closed current T. Next, we investigate some m-potential properties associated to T and we study a generalization of the Demailly and Xing results about the definition and the continuity of the complex hessian operator. We also prove a Xing-type comparison principle for the class . Finally, we generalize the work of Ben Messaoud–El Mir on the complex Monge–Ampere operator and the Lelong–Skoda potential associated to a positive closed current.
TL;DR: The connection between several hyperbolic type metrics is studied in subdomains of the Euclidean space with new metric introduced and compared to the distance ratio metric.
Abstract: The connection between several hyperbolic type metrics is studied in subdomains of the Euclidean space. In particular, a new metric is introduced and compared to the distance ratio metric.
TL;DR: In this paper, a wave front set suitable for the analysis of tempered ultradistributions of quasi-analytic Gelfand-Shilov type was introduced and the transformation properties of wave front sets were studied.
Abstract: We introduce a global wave front set suitable for the analysis of tempered ultradistributions of quasi-analytic Gelfand–Shilov type We study the transformation properties of the wave front set and use them to give microlocal existence results for pullbacks and products We further study quasi-analytic microlocality for classes of localization and ultradifferential operators, and prove microellipticity for differential operators with polynomial coefficients
TL;DR: In this article, a normal form for a class of real formal surfaces defined near a degenerate CR singularity is constructed, where the coefficients of and are vanishing, and a real-valued homogeneous polynomial in of degree such that the coefficients are vanishing.
Abstract: Let (z, w) be the coordinates in . We construct a normal form for a class of real formal surfaces defined near a degenerate CR singularity as followswhere is a real-valued homogeneous polynomial in of degree such that the coefficients of and are vanishing.
TL;DR: In this article, the theory of slice regular compositions is studied and the Littlewood subordination principle and the Denjoy-Wolff-type theorem are established for slice regular functions.
Abstract: In the article, we study the theory of slice regular compositions. Especially, the Littlewood subordination principle and the Denjoy–Wolff-type theorem are established for slice regular functions.
TL;DR: In this article, the authors studied vector mixed Sobolev-type partial differential equations (PDE) over octonions and developed a technique of their integration over the octonion algebra.
Abstract: Vector mixed Sobolev-type partial differential equations (PDE) are studied. The technique of their integration over the octonion algebra is developed. Theorems about integration of vector mixed Sobolev-type PDE over octonions are proved.
TL;DR: In this paper, the main aim of the present paper is to study a characterization of the class and some of its consequences, and the main focus of the paper is on the consequences of such a characterization.
Abstract: The main aim of the present paper is to study a characterization of the class and some of its consequences.
TL;DR: In this article, the existence of solutions for a class of p(x)-biharmonic equations with Navier boundary conditions was investigated based on variational methods and critical point theory.
Abstract: This paper deals with the existence of solutions for a class of p(x)-biharmonic equations with Navier boundary conditions. The approach is based on variational methods and critical point theory. Indeed, we investigate the existence of two solutions for the problem under some algebraic conditions with the classical Ambrosetti–Rabinowitz condition on the nonlinear term. Moreover, by combining two algebraic conditions on the nonlinear term which guarantee the existence of two solutions, applying the mountain pass theorem given by Pucci and Serrin we establish the existence of the third solution for the problem.
TL;DR: In this paper, the authors studied the Dirichlet problem for the Stokes resolvent system for bounded boundary data on bounded and unbounded domains with compact Lyapunov boundary.
Abstract: The paper studies the Dirichlet problem for the Stokes resolvent system for bounded boundary data on bounded and unbounded domains with compact Lyapunov boundary. (The boundary might be disconnected.) For a bounded domain, we prove the existence of a unique solution of the problem such that the velocity part is bounded. For an unbounded domain, we prove the existence of such a solution. But this solution is not unique. We characterize all solutions of the problem. Then we study bounded solutions of the nonlinear Dirichlet problem , in , on , where F is bounded. As a consequence, we study bounded solutions of the Dirichlet problem for the Darcy-Forchheimer-Brinkman system , . At last we prove a generalized maximum modulus principle for a solution of the Stokes resolvent system such that the velocity part is bounded.
TL;DR: In this article, the existence of weak solutions for the fractional p-Laplacian equation in the Caratheodory function is discussed and critical point theorems are discussed.
Abstract: In this paper, we use some critical point theorems to discuss the existence of weak solutions for the fractional p-Laplacian equation in where , , is a parameter, is the fractional p-Laplacian and is a Caratheodory function.
TL;DR: In this article, continuous and monotone curves for Dirichlet problems with p, q-Laplacians and indefinite nonlinearities are derived explicitly via minimax variational principle of the extended functional method.
Abstract: The paper is devoted to the study of two-parametric families of Dirichlet problems for systems of equations with p, q-Laplacians and indefinite nonlinearities. Continuous and monotone curves and on the parametric plane , which are the lower and upper bounds for a maximal domain of existence of weak positive solutions, are introduced. The curve is obtained by developing our previous work and it determines a maximal domain of the applicability of the Nehari manifold and fibering methods. The curve is derived explicitly via minimax variational principle of the extended functional method.
TL;DR: Fornaaess and Wiegerinck as mentioned in this paper extended this result by extending it to domains with boundaries locally given by graphs of continuous functions, and showed that the result holds even for domains with continuous functions.
Abstract: We extend a result by Fornaaess and Wiegerinck [Ark. Mat. 1989;27:257-272] on plurisubharmonic Mergelyan type approximation to domains with boundaries locally given by graphs of continuous functions.
TL;DR: In this article, a sufficient condition for having closed range is given for pseudoconvex, unbounded domains in the sense that the range is bounded by a constant number of vertices.
Abstract: A sufficient condition for to have closed range is given for pseudoconvex, unbounded domains in .
TL;DR: In this article, the authors considered Hardy space decomposition of where stands for the open unit disc, and is its boundary, and obtained the same result for the unit circle by using rational functions.
Abstract: In this paper we consider Hardy space decomposition of where stands for the open unit disc, and is its boundary. Hardy spaces decompositions for and for are, as classical results, available in the literature. For the basic tools are the Plemelj formula and the boundedness of the Hilbert transformation. For neither on the real line, nor on the unit circle, a Plemelj formula, or Hilbert transformation are available. In a recent paper Deng and Qian obtain Hardy spaces decomposition for on the real line by means of rational approximation. In the present paper by using rational functions, we achieve the same goal for for the range The work on the unit circle exposes the particular features of the kind of decomposition in the compact situation adaptable to higher dimensions.