We consider the Cauchy problem for an implicit system of linear partial differential equations. It belongs to the class of pseudohyperbolic systems. We prove the unique solvability of the Cauchy problem in weighted Sobolev spaces and obtain energy estimates of the solution.

In this paper, we consider nonlocal boundary value problems for the Laplace operator in a ball, which are a multidimensional generalisation of the Samarskii–Ionkin problem. The well-posedness of the problems are investigated, and Fredholm property of the problems are studied. Moreover, we obtain integral representations of their solutions in explicit forms.

ABSTRACT In this paper, we show Lyapunov and Hartman-Wintner-type inequalities for a fractional partial differential equations with Dirichlet conditions and we give some applications of these inequalities for the eigenvalue problem. Also, we give de La Vallée Poussin-type inequality for the fractional elliptic boundary value problem and Lyapunov-type inequalities for the fractional elliptic systems

In this note, we prove that the equal-area condition is necessary for the formation of internal transition layer by the solutions of a general singularly perturbed elliptic problem. Our result is independent of boundary conditions, it improves previous references in the literature and can be applied to Allen-Cahn, Sine-Gordon and Fisher-KPP equations, for example.

In this article, we study mappings acting between domains of two factor spaces by certain groups of Möbius automorphisms of the unit ball that act discontinuously and do not have fixed points. For such mappings, we have established estimates for the distortion of the modulus of families of paths, which are similar to the well-known Poletsky and Väisälä inequalities. As applications, we have obtained

In this paper, we consider the following Choquard equation: − Δ u + u = ( I α ∗ F ( u ) ) F ′ ( u ) i n   R N where N ⩾ 3 , α ∈ ( 0 , N ) , I α is the Riesz potential, and F ( u ) := 1 p | u | p + 1 q | u | q , where p = N + α N and q = N + α N − 2 are lower and upper critical exponents in sense of the Hardy–Littlewood–Sobolev inequality. Based on perturbation method and the invariant sets of descending

ABSTRACT We investigate the weak solvability and properties of weak solutions to the Dirichlet problem for a scalar elliptic equation − Δ u + b ( α ) ⋅ ∇ u = f in a bounded domain Ω ⊂ R 2 containing the origin, where f ∈ W q − 1 ( Ω ) with q>2 and b ( α ) := b − α x | x | 2 , b is a divergence-free vector field and α ∈ R is a parameter.

The existence of at least one positive radial solution of the generalized Kohn–Laplacian problem (1) − Δ H n u + R ( ξ ) u = ∑ i = 1 k a i ( | ξ | H n ) | u | p i − 2 u     − ∑ j = 1 m b j ( | ξ | H n ) | u | q j − 2 u     ξ ∈ Ω , u > 0 ξ ∈ Ω , ∂ u ∂ n = 0 ξ ∈ ∂ Ω , (1) is proved, where Δ H n is the Kohn–Laplacian (Heisenberg–Laplacian) operator, Ω is a Korányi ball, a i , 1 ≤ i ≤ k and b j , 1 ≤ j

The work is devoted to the holomorphy of a formal homogeneous series whose restrictions to analytic curves passing through 0 are holomorphic. In content, the main result of the paper is a variation of the well-known Forelli theorem on the holomorphic continuation of formal series along with a set of complex lines.

We study the Liouville-type theorems for the sub-elliptic inequality − Δ λ u ≥ u p i n   R N and for the corresponding system − Δ λ u ≥ v p − Δ λ v ≥ u q i n   R N , where p , q ∈ R , and Δ λ is a strongly degenerate operator of the form Δ λ = ∑ i = 1 N ∂ x i ( λ i 2 ∂ x i ) . Here the functions λ i ,   i = 1 , 2 , … , N , satisfy certain conditions such that Δ λ is homogeneous of degree 2 with respect

Based upon the method of moving spheres, we establish some Liouville-type results of a weighted elliptic equation. For subcritical nonlinearity, we prove non-existence result on the entire space. For supercritical nonlinearity, we obtain non-existence on bounded star-shaped domains. Meanwhile, a regularity result is also obtained.

We analyze Bergman spaces A f p ( D ) of generalized analytic functions of solutions to the Vekua equation ∂ ¯ w = ( ∂ ¯ f / f ) w ¯ in the unit disc of the complex plane, for Lipschitz-smooth non-vanishing real valued functions f and 1 < p < ∞ . We consider a family of bounded extremal problems (best constrained approximation) in the Bergman space A p ( D ) and in its generalized version A f p ( D

The real monotone decreasing sequence m 1 = 3 , m 2 = 2 3 , ( m 1 2 − 1 = 2 ) , m k + 2 = α k m k + 1 + β k β k m k + 1 + α k , α k = m k m k + 1 − 1 , β k = m k − m k + 1 , k ∈ N , converging to 1 is used to construct a parqueting of the complex plane leading to the harmonic Green function and the Poisson kernel for the countably many domains of the parqueting clustering at the point 1.

In this paper, we consider equations involving a class of nonlocal Monge–Ampère operators. We first establish some maximum principles, such as a narrow region principle and a decay at infinity. Then, by using the direct method of moving planes, we obtain the symmetry of solutions in an ellipsoid region and in the whole space. We also prove that the positive solutions are non-existence on the upper

In this article, we study the Schrödinger–Newton equations − Δ u + 2 π φ u = μ u + | u | p u , i n   R 2 , Δ φ = u 2 , i n   R 2 . We prove the existence of positive spherically symmetric decreasing solutions for p ∈ ( 0 , 2 ) by constrained minimization methods. For p = 2, by the Gagliardo–Nirenberg inequality, an elaborate estimate implies similar existence results. We also give the regularity, exponential

The parqueting-reflection principle is shown to work for constructing the harmonic Green functions and harmonic Neumann functions for a class of domains, which are bounded by two intersecting circular arcs in C ∞ with a particular angle π / n , n ∈ N ∗ . With the explicit expressions of the harmonic Green and Neumann functions, we solve the Dirichlet and Neumann boundary problems to the Poisson equation

Let D n , m = { ( z , w ) ∈ C n × C m : | w | 2 < e − μ | z | 2 } be the Fock–Bargmann–Hartogs domain, where μ > 0 . For any fixed positive integer m, we prove that there exists a unique positive integer n 0 such that D n , m is not a Lu Qi-Keng domain if and only if n ≥ n 0 , which verifies a conjecture posed by Yamamori. Meanwhile, we show that the Bergman projection is bounded on L p ( D n , m )

In this paper, we consider a nonlinear fractional equation with critical exponent on higher dimensional bounded domain. Due to the existence of critical points at infinity, this problem present a lack of compactness. To overcome this difficulty, we prove a version of Morse lemma at infinity for this problem. Then we characterize the critical points at infinity associated to this problem which allows

In this paper, we systematically study (slice regular) weighted composition operators on quaternionic Fock spaces. More precisely, the following results are obtained: Some relationship for weighted composition operators between the quaternionic and classical settings is explored to give criteria for boundedness and compactness of weighted composition operators, and to estimate the essential norm, Schatten

The paper aims to estimate the error of a cubature formula acting on an arbitrary function from the Sobolev space on a multidimensional cube. The norm of the error in the dual space of the Sobolev class is represented as a positive definite quadratic form in the weights of the cubature formula. Given a finite smoothness s of the Sobolev space, we establish a power-law convergence order to zero of error

We study asymptotically regular elliptic equations with L p ( x ) -logarithmic growth under the assumptions that the variable exponent satisfies strong log-Hölder continuity and the underlying domain is Reifenberg flat. To ensure a global Calderón–Zygmund estimate for such asymptotically regular problem, we make use of approximating the solutions for asymptotically regular problem by the solutions

We use the representation theory of nilpotent Lie groups of step two to find the fundamental solution for the Paneitz operator on the anisotropic quaternionic Heisenberg group. For the isotropic case, we also construct the closed-form of this solution. This result is the extension of the conclusion on the Heisenberg group.

In this paper, we consider the quasilinear elliptic problem with potential ( P ) − div ( φ ( x , | ∇ u | ) ∇ u ) + V ( x ) | u | q ( x ) − 2 u = f ( x , u ) in   Ω , u = 0 on   ∂ Ω , where Ω is a smooth bounded domain in R N ( N ≥   2 ), V is a given function in a generalized Lebesgue space L s ( x ) ( Ω ) , and f ( x , u ) is a Carathéodory function satisfying suitable growth conditions. Using variational

This work deals with the mth order elliptic equation with non-smooth coefficients in grand-Sobolev space generated by the norm of the grand-Lebesgue space L q ) ( Ω ) ,   1 < q < + ∞ . These spaces are non-separable, and therefore, to use classical methods for treating solvability problems in these spaces, you need to modify these methods. To this aim, we consider some subspace, where the infinitely

Let f be a nonconstant meromorphic function of finite order, and L ( z , f ) be a linear difference polynomial in f with small meromorphic coefficients. Some uniqueness theorems related to f sharing three values with L ( z , f ) are obtained. We also prove that f is identical with its difference operator Δ η n f , if they share three values CM. Some examples are given to show our results are best possible

The article is devoted to the study of linear inverse problems of finding, alongside the solution, also the unknown right-hand side for second-order differential equations. The method of the study is based on reducing the initial inverse problems to direct, already nonlocal, boundary value problems for loaded (integro-differential) equations, proving the solvability of the new problems and then constructing

Consider a class of modified Kirchhoff-type equations − ( 1 + b ∫ R 3 | ∇ u | 2 d x ) Δ u + V ( x ) u − 1 2 u Δ ( u 2 ) = f ( u ) , in   R 3 , where the nonlinear term f is 4-superlinear at infinity. By using the method of invariant sets of descending flow, the existence of a sign-changing solution is obtained. When f is assumed to be odd, we prove that the above problem admits infinitely many sign-changing

We study the linear stability of a resting state for a generalization of the basic rheological Pokrovski–Vinogradov model for flows of solutions and melts of an incompressible viscoelastic polymeric medium to the nonisothermal case under the influence of magnetic field. We prove that the corresponding linearized problem describing magnetohydrodynamic flows of polymers in an infinite plane channel has

The question on asymptotic representations of Green's function for classical boundary value problems for second-order elliptic equations depending on a complex parameter is considered. The equations are defined either in a domain with a compact boundary or in the whole space or the half-space. The main asymptotic term is written out. The results can be applied for the study of some inverse problems

In this paper we study weighted composition operators W g , ψ commuting with self-adjoint weighted composition operators W f , ϕ on the Hardy space H 2 . In particular, we investigate which combinations of symbol functions g and ψ give rise to W g , ψ commuting with a self-adjoint weighted composition operator on H 2 . We also show that these commuting weighted composition operators are normal.

Let p i ≥   2 and consider the following anisotropic p-Laplace equation − ∑ i = 1 N ∂ ∂ x i ∂ u ∂ x i p i − 2 ∂ u ∂ x i = g ( x ) f ( u ) , u > 0   i n   Ω . Under suitable hypothesis on the weight function g we present an existence result for f ( u ) = e 1 u in a bounded smooth domain Ω and nonexistence results for f ( u ) = − e 1 u or − ( u − δ + u − γ ) , δ , γ > 0 with Ω = R N respectively.

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