In this paper, we are concerned with the Hénon–Hardy type systems on R n : ( − Δ ) α 2 u ( x ) = | x | a v p ( x ) , u ( x ) ≥ 0 ,   x ∈ R n , ( − Δ ) α 2 v ( x ) = | x | b u q ( x ) , v ( x ) ≥ 0 ,   x ∈ R n , where n ≥ 2 , n > α , 0 < α ≤ 2 or α = 2 m . We prove Liouville theorems (i.e. non-existence of nontrivial nonnegative solutions) for the above Hénon–Hardy systems. The arguments used in our

We show that a family F = { f } of functions holomorphic in a domain Ω ⊂ C n is normal if all eigenvalues of the complex Hessian matrix of log ⁡ ( 1 + | f | 2 ) are uniformly bounded away from zero on compact subsets of Ω.

We deal with m-component vector-valued solutions to the Cauchy problem for a linear both homogeneous and nonhomogeneous weakly coupled second-order parabolic system in the layer R T n + 1 = R n × ( 0 , T ) . We assume that coefficients of the system are real and depending only on t, n ≥   1 and T < ∞ . The homogeneous system is considered with initial data in [ L p ( R n ) ] m , 1 ≤ p ≤ ∞ . For the

We extend to the singular case the results in [El Bahja H. Regularity for anisotropic quasi-linear parabolic equations with variable growth. Electron J Differ Equ. 2019;2019(104):1–21] concerning the regularity of weak solutions of a class of anisotropic quasi-linear parabolic equations with variable exponents. The proof is based on DiBenedetto's technique called intrinsic scaling; by choosing an appropriate

The aim of this paper is to give a proof of Zalcman–Pang's Rescalling Lemma in C n .

In this paper, we obtain new results of the Spanne and Adams type boundedness characterization of the fractional maximal commutator operator M b , α , 0 ≤ α < Q on the Carnot group G (i.e. nilpotent stratified Lie group) in generalized Morrey spaces M p , ϕ ( G ) , respectively, where Q is the homogeneous dimension of G .

In this paper we obtain the local regularity estimates in Besov spaces of weak solutions for a class of quasilinear elliptic equations d i v   a ∇ u ∇ u = d i v   f in   Ω . Moreover, we would like to point out that our results improve the known results for such equations.

Let X be a connected K-complete normal complex space. If for every closed discrete set A in X there exists a family F of holomorphic functions on X such that N ( F ) = A , then the K-envelope H ( X ) of holomorphy of X in the sense of Kerner (Math Ann. 1959;138:316–328) coincides with X.

In this paper, we deal with a class of Schrödinger–Kirchhoff problems in R N , driven by the fractional magnetic operator, involving critical Sobolev–Hardy nonlinearities and a nontrivial perturbation term. By variational approach, we obtain the existence of solutions which tend to zero under a suitable value of λ. The main feature and difficulty of our equations is the presence of the magnetic field

We prove Carleson embeddings for Bergman–Orlicz spaces of the unit ball that extend the lower triangle estimates for the usual Bergman spaces.

Let D p 1 , p 2 , … , p n = { z ∈ C n : ∑ j = 1 n | z j | p j < 1 } , p j > 1 , j = 1 , 2 , … , n . We mainly establish the sharp estimates of main coefficients of all homogeneous expansions for starlike mappings and starlike mappings of order α on D p 1 , p 2 , … , p n under some weak restricted conditions. Our derived results are the generalization of the corresponding classical results in one complex

We consider a parametric nonlinear Dirichlet problem driven by the p-Laplacian plus an indefinite potential. In the reaction, we have the competing effects of a parametric singular term and of a superlinear perturbation. We examine the nonexistence, existence and multiplicity of positive solutions as the parameter λ > 0 varies.

We prove the existence of infinitely many nonnegative solutions to the following nonlocal elliptic partial differential equation involving singularity ( − Δ ) p ( ⋅ ) s u = λ | u | γ ( x ) − 1 u + f ( x , u ) i n   Ω , u = 0 i n   R N ∖ Ω , where Ω ⊂ R N , N ≥ 2 is a smooth, bounded domain, λ > 0 , s ∈ ( 0 , 1 ) , γ ( x ) ∈ ( 0 , 1 ) for all x ∈ Ω ¯ , N > s p ( x , y ) for all ( x , y ) ∈ Ω ¯ × Ω ¯

In this study, we define double weighted variable exponent Sobolev spaces W 1 , q ( ⋅ ) , p ( ⋅ ) Ω , ϑ 0 , ϑ with respect to two different weight functions. Also, we investigate the basic properties of this spaces. Moreover, we discuss the existence of weak solutions for weighted Dirichlet problem of p ( ⋅ ) -Laplacian equation − d i v ϑ ( x ) ∇ f p ( x ) − 2 ∇ f = ϑ 0 ( x ) f q ( x ) − 2 f x ∈ Ω

In this paper, we consider a fractional p-Laplacian system with both concave–convex nonlinearities and sign-changing weight functions in bounded domains: ( − Δ ) p s u = λ f ( x ) | u | q − 2 u + 2 α α + β h ( x ) | u | α − 2 u | v | β i n   Ω , ( − Δ ) p s v = μ g ( x ) | v | q − 2 v + 2 β α + β h ( x ) | u | α | v | β − 2 v i n   Ω , u = v = 0 i n   R n ∖ Ω . By using the Nehari manifold, together

In this paper, we study hyponormality of the dilation of truncated Toeplitz operators on L 2 . In particular, we provide necessary and sufficient conditions for the dilation of truncated Toeplitz operators to be hyponormal. Moreover, we investigate connections between hyponormal and normal dilations of truncated Toeplitz operators.

We study the strong instability of ground state standing waves for the focusing L 2 -supercritical NLS with inverse-square potential i ∂ t u + Δ u + c | x | − 2 u = − | u | α u , ( t , x ) ∈ R × R d , where d ≥ 3 , u : R × R d → C , c ≠ 0 satisfies c < λ ( d ) := ( d − 2 ) / 2 2 and 4 / d < α < 4 / ( d − 2 ) . This result extends a recent result of Bensouilah et al. [On stability and instability of

In this paper, we study the Schrödinger equation − Δ u = f ( u ) + h ( x ) i n   R N , where N ≥ 3 , h ( x ) ≢ 0 and f satisfies the conditions of Berestycki-Lions type. On the basis of variational methods, by using Ekeland's variational principle, the mountain pass theorem and a Pohožaev type identity, we obtain the existence of ground state solutions and the multiplicity of positive solutions under

We study a Dirichlet type problem for an equation involving the fractional Laplacian and a reaction term subject to either subcritical or critical growth conditions, depending on a positive parameter. Applying a critical point result of Bonanno, we prove existence of one or two positive solutions as soon as the parameter lies under an (explicitly determined) value. As an application, we find two positive

The paper deals with the following Kirchhoff equations: − a + b ∫ R N | ∇ u | 2 d x △ u + V ( x ) u = K ( x ) f ( u )   i n   R N , u ∈ H 1 ( R N ) , where N ≥ 3 , f ( u ) is C 1 real function satisfying quasicritical growth at infinity, and V ( x ) , K ( x ) are positive and continuous functions. Combining Mountain Pass Theorem and compact embeddings in weighted Sobolev spaces, we establish the existence

In this paper, we study the existence of nodal solutions for the Kirchhoff-type problem with concave-convex nonlinearities − a + b ∫ Ω | ∇ u | 2 d x Δ u = λ | u | q − 1 u + | u | p − 1 u i n   Ω , u = 0 o n   ∂ Ω , where Ω is a bounded domain with smooth boundary ∂ Ω in R N ( N = 1 , 2 , 3 ) , a 0 < q < 1 ,   3 < p < 5 . By constraining the energy functional of the above problem on a subset of the

In this present paper, we study positivity and completeness of some invariants metrics on unbounded domains in C n and generalizations of some theorem in complex spaces.

The prime purpose of this article aims to establish connections between various sub-classes of harmonic univalent mappings by applying a convolution operator associated with the well-known Wright function. In particular, the inclusion relations among Goodman–Rønning-type harmonic univalent mappings, harmonic k-uniformly convex mappings and harmonic k-uniformly starlike mappings in the plane are established

The goal of this work is to extend the concepts of generalized Lelong number of positive currents investigated by Skoda, Demailly and Ghiloufi in complex analysis, to weakly positive supercurrents on the real superspaces. We generalize then a result of Lagerberg when the supercurrent is closed as well as a very recent result of Berndtsson for minimal supercurrents associated to submanifolds of R n

The paper is devoted to the proof of new estimations of harmonic measure of a boundary arc of a Jordan domain D. The main result allows to estimate a distortion of harmonic measure under locally quasiconformal or quasiconformal mappings of the unit disk of complex plane onto domain D. As an application of the main results, some estimations for quasiconformal mappings of the disk are obtained.

In this paper we show that continuous endomorphisms on the space of entire functions with a given order can be expressed as differential operators of infinite order satisfying suitable growth conditions.

For any Lipschitz domain we construct an arbitrarily small, localized perturbation which splits the spectrum of the Dirichlet, Neumann or Robin Laplacian into simple eigenvalues. We showcase two different approaches. The first one consists in the excision of a hole inside the domain and the perturbation of its boundary, and is based on a Hadamard's formula and sharp spectral stability estimates. The

This paper concerns univalent analytic functions f for which log⁡f′ belong to Dirichlet-Morrey spaces. We establish some characterizations of Dirichlet-Morrey domains. Our results involve Schwarzian derivatives, quasiconformal extensions and Grunsky kernels of univalent functions.

The n-dimensional generalized Hartogs triangles are domains defined by Hpn:={(z1,…,zn)∈Cn:|z1|p1<⋯<|zn|pn<1} with p:=(p1,…,pn)∈(Z+)n and n≥2. In this paper, we first obtain an estimate for the Bergman kernel of Hpn and then use it to establish the Lp boundedness of the associated Bergman projections. Our result generalizes the Lp boundedness result for two-dimensional generalized Hartogs triangles

In this paper, we consider a type of Kirchhoff problem as follows −(a+b∫RN|∇u|2)Δu=(1+εK(x))u2∗−1,u>0, in RN, where a, b>0 are given constants, ε>0 is a small parameter and 2∗=2N/(N−2),(N≥3). We show that if K(x) has k critical points near which K(x) satisfies some expansion assumption, then by Lyapunov–Schmidt reduction method, we construct multi-peak solutions for ε>0 small, which concentrate at

We completely solve several important problems, such as boundedness and compactness, topological structure, ergodic and dynamical properties, for composition operators on weighted Banach spaces of entire functions with sup-norm.

In this paper, we consider the Choquard equations with a local perturbation −Δu+u=∫RN|u(y)|2|x−y|N−αdyu+|u|q−2uin RN, where N≥3, α∈((N−4)+,N), q∈(2,2N/(N−2)). By using variational method and approximating approach, we prove that for any given positive integer k, the above equation has a least energy radial solution changing sign exactly k times. This solution is constructed as the limit of such solutions

Let M be a complete Kähler manifold, whose universal covering is biholomorphic to a ball Bm(R0) in Cm ( 02n+1+α+ρK) hyperplanes in general position regardless of multiplicity with certain positive constants K and α<1 (explicitly estimated), then f1=f2 or f2=f3 or f3=f1. Moreover, if the above three mappings share the hyperplanes with mutiplicity counted to level n + 1 then f1=f2=f3. Our results generalize

In this paper, we study the existence of positive solutions to a class of singular quasilinear elliptic problems −div(|x|−ap|∇u|p−2∇u)+V(x)|u|p−2u=λh(x)|u|q−2u+H(x)|u|m−2u,x∈RNu(x)>0,x∈RN,u∈X≡Da1,p(RN,|x|−ap)∩Lp(RN,V), where 10,1

In this paper, after introducing the concepts of quaternionic dual group and the quaternionic valued character on locally compact abelian group G2, the inverse of the quaternionic Fourier transform (QFT) on locally compact abelian groups is investigated. Due to the non-commutativity of multiplication of quaternions, there are different types of QFTs right, left and two-sided quaternionic Fourier transform

We study polynomial interpolation of Hermite type of holomorphic functions based on Radon projections. We give two kinds of interpolation schemes and show that the interpolation polynomials are continuous with respect to the angles and the distances. When the chords are suitably distributed, we prove that the interpolation polynomials converge geometrically on the closed unit disk to the functions

In this paper, we characterize the kernels and finite rank properties of the Hankel operators and small Hankel operators with pluriharmonic symbols on Hardy-Sobolev spaces Hβ2 and pluriharmonic Hardy-Sobolev spaces Hβ,ph2. For β=12, the result was given by Y. J. Lee.

(2020). Professor Jinyuan Du: Citation for his 70th birthday. Complex Variables and Elliptic Equations. Ahead of Print.

In this paper, the authors study the boundedness of the Toeplitz type operator Tϕ and the Hankel type operator Vϕ on Hp,q,s(B). Further, the authors discuss the solvability of the Gleason's problem on Hp,q,s(B). Finally, the inclusion relations of the space Hp,q,s(B) and some typical function spaces are also considered.

We present an Almansi-type decomposition for polynomials with Clifford coefficients, and more generally for slice-regular functions on Clifford algebras. The classical result by Emilio Almansi, published in 1899, dealt with polyharmonic functions, the elements of the kernel of an iterated Laplacian. Here, we consider polynomials of the form P(x)=∑k=0dxkak, with Clifford coefficients ak∈Rn, and get

In this paper, we consider a class of fractional Laplacian problems of the form: (−Δ)p1(x,.)su+(−Δ)p2(x,.)su+|u|q(x)−2u=g(x)u−γ(x)+λf(x,u)in Ω,u=0,on ∂Ω, where Ω⊂RN, (N≥2), is a bounded domain and (−Δ)pi(x,.)s is the fractional pi(x,.)-Laplacian. We assume that λ is a nonnegative parameter and γ:Ω¯→(0,1) is a continuous function. By using variational methods and monotonicity arguments combined with

In this article, we study the existence/multiplicity results for the variable order nonlocal Choquard problem with variable exponents (−Δ)p(⋅)s(⋅)u(x)=λ|u(x)|α(x)−2u(x)+∫ΩF(y,u(y))|x−y|μ(x,y)dyf(x,u(x)),x∈Ω,u(x)=0,x∈Ωc:=RN∖Ω, where Ω⊂RN is a smooth and bounded domain, N≥2, p,s,μ and α are continuous functions on RN×RN and f(x,t) is a Carathéodory function with F(x,t):=∫0tf(x,s)ds. Under suitable assumption

The boundedness, invertibility and smoothness properties of multidimensional singular integral operators and solvability of the corresponding singular integral equations are investigated in Besov spaces.

A model of resonators for which boundary is composed of the Helmholtz resonators is constructed. It is based on the theory of self-adjoint extensions of symmetric operators. We consider a limiting procedure when the number of resonators tends to infinity and look after the spectrum of the Neumann Laplacian. It is shown that there is a sequence of the model systems and, correspondingly, a sequence of

Answering a question about triangle inequality suggested by R. Li, Barrlund [The p-relative distance is a metric. SIAM J Matrix Anal Appl. 1999;21:699.702] introduced a distance function which is a metric on a subdomain of Rn. We study this Barrlund metric and give sharp bounds for it in terms of other metrics of current interest. We also prove sharp distortion results for the Barrlund metric under

We solve the Riemann boundary value problem for bi-analytic functions on a contour consisting of non-rectifiable closed Jordan curves. In the classical results on this problem, curves are piecewise-smooth. But the Riemann boundary value problem has a lot of applications in the theory of solid media and other fields, and some of these applications allow fractal (and, consequently, non-rectifiable) contours

The aim of the D-bar method is to solve inverse conductivity problems. For complex conductivity, Hamilton et al. presented a reconstruction method which consists of six steps. In this article, we present a new regularization denoted by Sm based on an iterative algorithm which allows us to reconstruct an approximation of the conductivity of the domain. The advantage of this new regularization is the

This paper is concerned on the fourth-order elliptic equation Δ2u−Δu+V(x)u−λΔ[ρ(u2)]ρ′(u2)u=f(u) in RN,u∈W2,2(RN),Pλ where Δ2=Δ(Δ) is the biharmonic operator, 3≤N≤ 6, the radially symmetric potential V may change sign and infRNV(x)=−∞ is allowed. If f satisfies a type of nonquadracity and monotonicity conditions and ρ is a suitable smooth function, we prove, via variational approach, the existence

In the present paper, by using a variational approach combined with the Nehari manifold method and fibreing maps, the existence of two non-trivial solutions for a class of p-biharmonic problems is obtained.

We make sharp estimates to obtain a Schwarz-type lemma for the symmetrized polydisc Gn and for the extended symmetrized polydisc G~n. We explicitly construct an interpolating function under certain condition. To do so, we followed the methods described in [Young NJ. The automorphism group of the tetrablock. J Lond Math Soc. 2008;77(2):757–770]. Also we find a few geometric interplay between the members

In this article, we introduce a subclass B(α,μ,A,B) of Bazilevič functions of order α+iμ and estimate the logarithmic coefficients and the difference of moduli of successive coefficients for the function belonging to the above class. The results obtained generalize some known results.

In this paper, we study the existence of a positive solution to the elliptic problem: (−Δ)su=u−q+f(x)h(u)+μin Ω,u>0in Ω,u=0in (RN∖Ω). Here Ω⊂RN (N>2s) is an open bounded domain with smooth boundary, s∈(0,1) and q∈(0,1). For s∈(0,12), we take advantage of the convexity of Ω. The operator (−Δ)s indicates the restricted fractional Laplacian, and μ is a non-negative Radon measure as a source term. The

In this paper, we introduce a class of holomorphic functions Mg(U) on unit disk U. Let w(ξ) be a normalized holomorphic function on U such that w(ξ)/w′(ξ)∈Mg(U), and w(ξ)−ξ has a zero of order k + 1 at ξ=0. By using the formula of Faà di Bruno for the kth derivative of a composite function, we establish the Fekete and Szegö inequality for w(ξ), and then we extend this result to higher dimensions. The

This paper presents the global Borel-Pompieu- and the global Cauchy-type integral formulas for the quaternionic slice regular functions using the relationship between this function space and a non-constant coefficient differential operator given by G:=∥x→∥2∂0+x→∑k=13xk∂k, according to [González-Cervantes JO. On cauchy integral theorem for quaternionic slice regular functions. Complex Anal Oper Theory

In this paper, we are interested in the existence of solutions to the anisotropic nonlocal problem {(P)δ}−∑i=1N∂∂xi∂u∂xipi−2∂u∂xi=∫ΩF(x,u)rf(x,u)+δ|u|p∗−2uin Ω,u≥0in Ω,u=0on ∂Ω, where Ω is a smooth bounded domain of RN, N≥2, δ=0 or δ=1,pi and r≥0, 1

In this paper, we discuss several results regarding existence, multiplicity and stability of solutions of differential equation with discontinuous nonlinearities relevant in several applications. Our problem can be formulated as a free boundary problem with unknown boundaries to be characterized. We first show that the bifurcation diagram is a generalized ‘S-shaped’ curve. Then we study the stability

In this article, we study the quasilinear elliptic equation: −Δu+V(x)u−Δ[(1+u2)1/2]u2(1+u2)1/2=h(u),x∈RN, where N≥3, V:RN→R satisfies suitable assumptions. Unlike h∈C1(R,R), we only need to assume that h∈C(R,R). By using a change of variable, we obtain the existence of ground state solutions with general critical growth.

In this paper, 8×8 real matrix representations of elliptic biquaternions are obtained and by means of these representations, a general method is developed to solve the linear elliptic biquaternion equations. Then, this method is applied to the well-known quaternion equations X−QXR = S and QX−XR = S over the elliptic biquaternion algebra. Also, some illustrative numerical examples are given to show

The goal of this paper is to investigate a parametric Dirichlet problem with (p,q)-Laplacian and concave–convex nonlinearity. Denoting by Sλ the set of positive solutions of the problem corresponding to the parameter λ>0, we prove that the set Sλ is compact in the W01,p(Ω)-topolgy. We also establish an upper semicontinuity and a Mosco convergence result for the multivalued mapping λ↦Sλ.

In this article, we are concerned with the existence of solutions for the following critical fractional p&q-Laplacian system (1) (−Δ)ps1u+(−Δ)qs2u=λQ(x)|u|r−2u+2αα+β|u|α−2u|v|β,in Ω;(−Δ)ps1v+(−Δ)qs2v=μQ(x)|v|r−2v+2βα+β|u|α|v|β−2vin Ω;u=v=0,in Rn∖Ω,(1) where Ω is a smooth bounded set in Rn, λ,μ>0 are two parameters, 0ps1, and α>1,β>1 satisfy α+β=ps1∗ with ps1∗=npn−ps1 is the fractional Sobolev critical