ABSTRACT In Koo and Wang (Joint Carleson measure and the difference of composition operators on A α p ( B n ) . J Math Anal Appl. 2014;419:1119–1142), the authors introduced a concept of joint Carleson measure and used it to characterize when the difference of two composition operators on weighted Bergman space over the unit ball is bounded or compact. In this paper, we extend the concept of joint

ABSTRACT This paper generalizes to the context of smooth metric measure spaces and submanifolds with negative sectional curvatures some well-known geometric estimates on the p-fundamental tone by using vector fields satisfying a positive divergence condition. Choosing the vector field to be the gradient of an appropriately chosen distance function yields generalised McKean estimates whilst other choices

We consider the swallowtail integral Ψ ( x , y , z ) := ∫ − ∞ ∞ e i ( t 5 + x t 3 + y t 2 + z t ) d t for large values of | z | and bounded values of | x | and | y | . The integrand of the swallowtail integral oscillates wildly in this region and the asymptotic analysis is subtle. The standard saddle point method is complicated and then we use the modified saddle point method introduced in López et al

In this paper, we investigate the multiplicity of solutions for Kirchhoff fractional p-Laplacian system in bounded domains: ∑ i = 1 k [ u i ] s , p p θ − 1 ( − Δ ) p s u j ( x ) = λ j f j ( x ) | u j | q − 2 u j + ∑ i ≠ j β i , j h ( x ) | u i | m | u j | m − 2 u j in   Ω , u j = 0 i n   R n ∖ Ω . By using the Nehari manifold method, together with Ekeland's variational principle, we show that there

ABSTRACT There have been many extensions of the Nevanlinna's five-values theorem for meromorphic functions to the case of meromorphic maps into P n ( C ) . We improve these results by considering the degenerate case and using the weaker condition ‘ f − 1 ( H ) ⊆ g − 1 ( H ) ’ instead of the usual one ‘ f − 1 ( H ) = g − 1 ( H ) ’ for some hyperplanes H among the given hyperplanes. Our main theorems

ABSTRACT The triangular ratio metric is studied in a domain G ⊊ R n , n ≥ 2 . Several sharp bounds are proven for this metric, especially in the case where the domain is the unit disk of the complex plane. The results are applied to study the Hölder continuity of quasiconformal mappings.

This paper deals with the existence and nonexistence of positive radial solutions of the elliptic equation − Δ u = f ( | x | ,   u ) , u ∈ Ω , u | ∂ Ω = 0 ,   where Ω = { x ∈ R N : r 1 < | x | < r 2 } ,   N ≥ 3 and f ∈ C ( [ r 1 , r 2 ] × R + ) , whose sign may change. We present new eigenvalue criteria for the existence and nonexistence to this problem. The discussion is based on the fixed point index

In this paper, we consider the Choquard-type equation with the fractional Laplacian ( − Δ ) s u ( x ) + a u ( x ) = 1 | x | n − α ∗ u p u p − 1 , x ∈ R n , u ( x ) > 0 , x ∈ R n , where 0

We study the existence and asymptotic behavior of least energy sign-changing solutions to a gauged nonlinear Schrödinger equation with critical exponential growth − Δ u + ω u + λ h 2 ( | x | ) | x | 2 + ∫ | x | ∞ h ( s ) s u 2 ( s ) d s u = f ( u ) i n     R 2 , u ∈ H r 1 ( R 2 ) , where ω , λ > 0 are constants and h ( s ) = ∫ 0 s r 2 u 2 ( r ) d r . Under some suitable assumptions on f ∈ C ( R )

ABSTRACT We give the boundedness of the Hardy-Littlewood maximal operator M λ , λ ≥ 1 , on central Herz-Morrey-Musielak-Orlicz spaces H Φ , q , ω ( X ) over bounded non-doubling metric measure spaces and to establish a generalization of Sobolev's inequality for Riesz potentials I α , τ f , τ ≥ 1 , α > 0 , of functions in such spaces. As an application and example, we obtain the boundedness of M λ and

ABSTRACT In this paper we derive a Cauchy integral formula for holomorphic and hyperholomorphic functions in domains bounded by a Koch snowflake in two- and three-dimensional setting.

In this paper we study the first-order Hamiltonian system z ˙ = J H z ( t , z ) . Moreover precisely, assuming that the nonlinearity satisfies a local super-quadratic condition, which is weaker than the usual global super-quadratic condition, we obtain new existence results of ground state homoclinic orbits and infinitely many geometrically distinct homoclinic orbits by using a variational method.

ABSTRACT In this paper, we establish the existence of at least one positive solution to a singular elliptic equation with zero boundary data and critical Hardy–Sobolev exponent. We show that the existence of the positive solution in high dimension depends on the sign of the mean curvature of the boundary near zero and in the low dimensions, it depends on the mass of the domain.

In this work, we establish existence of weak solutions for quasilinear Schrödinger equations where the potential is bounded from below and above by positive constants. The nonlinearity has an iteration with higher eigenvalues for the associated linear problem. Hence, the energy functional associated with our main problem admits a linking structure. The main difficulty here comes from the fact that

In this paper, the monotonicity of solutions for fully nonlinear fractional order equations is studied. We establish a narrow region principle in bounded domains. Then using the sliding method, we obtain the monotonicity of the solutions of fully nonlinear fractional order equations on both bounded domains and the whole space.

In this paper, we deal with the variable growth Schrödinger–Poisson Systems in R 3 : − d i v ( | ∇ u | p ( x ) − 2 ∇ u ) + ( V ( x ) + K ( x ) ϕ ( x ) ) | u | p ( x ) − 2 u = f ( x , u ) , x ∈ R 3 , − Δ ϕ ( x ) = K ( x ) | u | p ( x ) , x ∈ R 3 , u ∈ W 1 , p ( x ) ( R 3 ) , where p ( x ) ≤ p ∗ ( x ) := 3 p ( x ) 3 − p ( x ) and V ( x ) , K ( x ) and f ( x , u ) are periodic in x. We use the non-Nehari

ABSTRACT In this paper we investigate solutions to the characteristic Goursat problem with analytic data for a class of linear operators of second order without the Levi's generalized condition. We give an explicit representation of the solutions of the problem involving Kummer functions and show that they are generally singular on the surface K tangent to the characteristic surface S.

ABSTRACT Let ( − Δ p ) s , with 0 < s < 1 < p < ∞ , be the fractional p-Laplacian operator. We prove that the span of p-harmonic functions in B 1 is dense in C k ( B ¯ 1 ) .

In the half-plane, the Dirichlet problem is considered for elliptic differential-difference equations with nonlocal general-kind potentials, which are linear combinations of translations of the desired function, not bounded by commensurability conditions. We find a condition for the symbol of the corresponding differential-difference operator, providing the classical solvability of the specified problem

ABSTRACT We deal with anisotropic singular perturbation problems in some weighted spaces of Sobolev type. The perturbed problems are elliptic, semi-linear and non-local. Using a variational approach, we establish the existence of their solutions as critical points of some C 1 -functionals. Besides, we study the asymptotic behaviour of these solutions with respect to the parameter of perturbation ε

Ingham-type results are the characterization of the existence of nonzero functions defined on the real line vanishing on a nonempty open set in terms of the decay of their Fourier transforms. In this paper, we prove analogous results for the Dunkl Fourier transforms.

In this paper we prove the nonexistence of nontrivial solutions to − Δ u = f ( u ) in  Ω , u = 0 on  ∂ Ω , with Ω ⊂ R N ( N ≥ 1 ) a bounded domain and f locally Lispchitz with nonpositive primitive. As a consequence, we discuss the long-time behavior of solutions to the so-called sine-Gordon equation.

ABSTRACT In this paper we establish the existence of solutions for Schrödinger equations involving fractional p-Laplacian with supercritical growth and nonlinearities indefinite in sign, by using variational methods combined with sub-super solutions arguments.

ABSTRACT We consider smooth deformations of the CR structure of a smooth 2-pseudoconcave compact CR submanifold M of a reduced complex analytic variety X outside the intersection D ∩ M with the support D of a Cartier divisor of a positive line bundle F X . We show that nearby structures still admit projective CR embeddings. Special results are obtained under the additional assumptions that X is a projective

We study the behavior of the boundary function of a harmonic mapping from global and local points of view. Results related to the Koebe lemma are proved, as well as a generalization of a boundary behavior theorem by Bshouty, Lyzzaik and Weitsman. We also discuss this result from a different point of view, from which a relation between the boundary behavior of the dilatation at a boundary point and

ABSTRACT In this paper we shall characterize the space of Fourier ultra-hyperfunctions by the Mehler kernel method. That is, we show that any Fourier ultra-hyperfunctions are characterized as initial values of the solutions of Hermite heat equation.

ABSTRACT The local unique solvability of the Cauchy-type problem to a semilinear equation in a Banach space, which is solved with respect to the highest order Riemann–Liouville derivative, is proved. A linear unbounded operator at the unknown function in the equation generates an analytic in a sector resolving the family of operators of the linear homogeneous fractional-order equation. This result

ABSTRACT In this paper, we give a correct definition of the Laplace operator with delta-like potentials. Correctly solvable pointwise perturbation is investigated and formulas of resolvent are described. We study some properties of the resolvent. In particular, we prove Krein's formula for these resolvents.

Let D be a bounded domain in C n , n ≥ 1 . In this paper, we study two biholomorphic invariants on D, the Fridman invariant e D ( z ) and the squeezing function s D ( z ) . More specifically, we study two questions about the quotient invariant m D ( z ) = s D ( z ) / e D ( z ) : (1) If m D ( z 0 ) = 1 for some z 0 ∈ D , is D biholomorphic to the unit ball? (2) Is m D ( z ) constantly equal to 1? We

For the stationary convection–diffusion equation in R n , n>2, we give a sufficient condition on the bounded non-compactly supported drift that guarantees the existence of a fundamental solution with global two-sided estimates of the Newtonian potential type.

ABSTRACT The work deals with the existence of solutions of an integro-differential equation in the case of the normal diffusion and the influx/efflux term proportional to the Dirac delta function. The proof of the existence of solutions is based on a fixed point technique. Solvability conditions for non Fredholm elliptic operators in unbounded domains are used.

We consider classes of harmonic univalent functions f k = h k + g k ¯ , ( k = 1 , 2 ) that are shears of the analytic map h k − g k = 1 / 2 log ⁡ [ 1 + z / 1 − z ] with dilatation ω k = e i θ k z k . We prove that if the convolution f 1 ∗ f 2 is locally one-to-one and sense-preserving, then f 1 ∗ f 2 is convex in the direction of the real axis.

Let C n be the space of n-dimensional complex variables and D n be the unit polydisc in C n . We obtain the distortion theorems of the Fréchet-derivative type and the Jacobi-determinant type for a certain subclass of normalized biholomorphic mappings defined on D n . Also, the distortion theorem of Jacobi-determinant type for the corresponding subclass defined on the unit ball in C n with arbitrary

The existence of a nontrivial solution in some ‘weaker’ sense of the system of equations: L s u + l ( x ) ϕ u + w ( x ) u k = μ i n   Ω , ( − Δ ) s ϕ = l ( x ) u 2 i n   Ω , u > 0 i n   Ω , u = ϕ = 0 i n   R N ∖ Ω is proved. Here, L s u = ( − Δ ) s u − λ u − β , λ > 0 , 0 < β ≤ s < 1 , l, w are bounded, nonnegative real-valued functions in Ω, μ is a nonnegative Radon measure and k>1 belongs to a certain

ABSTRACT We consider the 3-D Dirac operator with variable regular magnetic and electrostatic potentials, and singular potentials (1) D A , Φ , Q sin u ( x ) = D A , Φ + Q sin u ( x ) , x ∈ R 3 (1) where (2) D A , Φ = ∑ j = 1 3 α j i ∂ x j + A j ( x ) + α 0 m + Φ ( x ) I 4 , (2) Q sin = Γ ( s ) δ Σ is the singular potential with Γ ( s ) = Γ i j ( s ) i , j = 1 4 being a 4 × 4 matrix and δ Σ is the delta-function

ABSTRACT In view of recent developments of the study of reproducing kernel Hilbert spaces, in particular with the context the Hardy spaces on tubes, aspects of rational approximation for functions of finite energy in several complex and several real variables are developed.

ABSTRACT In this paper, boundedness, noetherity and smoothness properties of multidimensional singular integral operators and solvability of the corresponding singular integral equations in Besov spaces are studied.

This paper is concerned with the existence of positive solutions to Schrödinger-Kirchhoff-type equations ( P ) − a + b ∫ R 3 | ∇ u | 2 Δ u + V ( x ) u = | u | p − 1 u i n   R 3 , u ∈ H 1 ( R 3 ) , where a and b are two positive constants, p ∈ ( 1 , 5 ) and V : R 3 → R is a potential function. Under certain assumptions on V, we prove that ( P ) has no ground state solution. However, we can show ( P

In this paper, we are interested in existence and multiplicity of solutions to anisotropic elliptic equations of Kirchhoff-type given by L ( u ) = λ ∫ Ω F ( x , u ) r f ( x , u ) + | u | p ∗ − 2 u  in    Ω , u = 0  on    ∂ Ω , ( P ) λ where L ( u ) := − ∑ i = 1 N M ∫ Ω ∂ u ∂ x i p i ∂ ∂ x i × ∂ u ∂ x i p i − 2 ∂ u ∂ x i , Ω is a smooth bounded domain of R N , N > 2 , λ > 0 and r ≥ 0 are parameters

We show that the inverse problem of Calderon for conductivities in a two-dimensional Lipschitz domain is stable in a class of conductivities that are Dini continuous. This extends previous stability results when the conductivities are known to be Hölder continuous.

We investigate overdetermined problems for p-Laplace and generalized Monge–Ampére equations. By using the theory of domain derivative, we find duality results and characterization of the overdetermined boundary conditions via minimization of suitable functionals with respect to the domain.

We prove Poletskii-type moduli inequalities for the two-index scale of weighted bounded ( q , p ) -distortion under minimal regularity. This implies, in particular, a positive solution to a question formulated in a Tengval's paper on the validity of Poletskii-type moduli inequalities for nonspherical condensers, for mappings of Sobolev classes with the least possible summability exponent.

This paper is concerned with the existence and multiplicity of solutions for the fractional variable order Choquard type problem ( − Δ ) p ( ⋅ ) s ( ⋅ ) u ( x ) + ( − Δ ) q ( ⋅ ) s ( ⋅ ) u ( x ) = λ | u ( x ) | β ( x ) − 2 u ( x ) + ∫ Ω F ( y , u ( y ) ) | x − y | μ ( x , y ) d y f ( x , u ( x ) ) + k ( x ) i n   Ω , u ( x ) = 0 i n   R N ∖ Ω , where ( − Δ ) p ( ⋅ ) s ( ⋅ ) and ( − Δ ) q ( ⋅ ) s (

ABSTRACT Let X be a Riemannian n-dimensional smooth closed manifold, n ≥ 2 , E i be smooth vector bundles over X and { A i , E i } be an elliptic differential complex of linear first order operators. We consider the operator equations, induced by the Navier-Stokes type equations associated with { A i , E i } on the scale of anisotropic Hölder spaces over the layer X × [ 0 , T ] with finite time T>0

This article deals with mixed boundary value problem for an elliptic differential-difference equation in a cylinder. There are obtained results on the smoothness of generalized solutions of such problem in subdomains, and necessary and sufficient conditions for the preservation of smoothness on the boundaries of neighboring subdomains.

This paper is devoted to the study of the qualitative properties of solutions to boundary-value problems for strongly elliptic differential-difference equations. In contrast to elliptic differential equation, the smoothness of generalized solutions of boundary-value problems for differential-difference equations can be violated near the boundary of these subdomains even for infinitely differentiable

We consider the Helmholtz problem with source term in an anisotropic domain of R 3 . The aim of this paper is to investigate the interplay between the geometry and analysis of elliptic equations under small perturbation of domain. The solving of this problem, anisotropic as well as isotropic case, is based on integral equations. We exhibit the Lippmann-Schwinger integral equation in the presence of

ABSTRACT We consider the following Kirchhoff type equation: − a + b ∫ Ω | ∇ u | 2 d x Δ u = g ( u ) , i n   R 4 , where a, b are positive constants and g ∈ C ( R , R ) . Under the critical growth assumptions on g, we obtain the existence of least energy solutions by studying the associated minimization problem with a new constraint. The mountain pass characterization of least energy solutions is also

We generalize some earlier extensions of the modular inequality of Poleckii, which is a basic tool in the geometric theory of mappings. Used first in the well-known theory of quasiregular mappings, it was successfully used also for more general classes of mappings. Indeed, the mappings satisfying a Poleckii's modular inequality are the usual examples in the class of the mappings satisfying generalized

In this work, we investigate the second-order nonlinear elliptic system defined on an unbounded domain and with variable nonconstant coefficients. We establish the existence of the solution of the nonlinear problem for a second-order nonstrongly elliptic system by using a coercive estimate, which is obtained for the linear case. The nonlinear elliptic system is transformed into an equivalent fixed

ABSTRACT We consider the second-order semilinear elliptic system Δ u = p ( x ) v α , Δ v = q ( x ) u β , where x ∈ R N , N ≥ 3 , α and β are positive constants, p and q are nonnegative continuous functions. We prove that nontrivial nonnegative entire solutions fail to exist if the functions p and q are of slow decay.

ABSTRACT From a unique viewpoint, singular elliptic and parabolic second-order inequalities with quasilinear KPZ-type terms are investigated in cylindrical domains. The weight Neumann condition is set on the lateral area of the cylinder; no condition is set on the base of the cylinder (regardless the type of the equation). Results of two kinds are established: the existence of a limit of each solution

The aim of this paper is to find inequalities for 3/2 moments of the negative eigenvalues of Schrödinger operators on half-line that have a ‘Hardy term’ by using the commutator method.

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.