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Joint Carleson measure for the difference of composition operators on the polydisks Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2021-01-25 Hyungwoon Koo; Inyoung Park; Maofa Wang
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
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Geometric estimates on weighted p-fundamental tone and applications to the first eigenvalue of submanifolds with bounded mean curvature Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2021-01-21 Abimbola Abolarinwa; Ali Taheri
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
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The swallowtail integral in the highly oscillatory region III Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2021-01-20 Chelo Ferreira; José L. López; Ester Pérez Sinusía
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
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Multiple solutions for a coupled Kirchhoff system with fractional p-Laplacian and sign-changing weight functions Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2021-01-20 Maoding Zhen; Meihua Yang
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
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Improvement of the uniqueness theorems of meromorphic maps of ℂm into ℙn(ℂ) Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2021-01-14 Kai Zhou; Lu Jin
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
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Triangular ratio metric in the unit disk Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2021-01-14 Oona Rainio; Matti Vuorinen
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.
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Positive radial solutions for elliptic equations with sign-changing nonlinear terms in an annulus Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2021-01-14 Yonghong Ding; Yongxiang Li
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
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Symmetry and monotonicity of positive solutions for a Choquard equation with the fractional Laplacian Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2021-01-13 Xiaoshan Wang; Zuodong Yang
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
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Sign-changing solutions to a gauged nonlinear Schrödinger equation with critical exponential growth Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2021-01-12 Liejun Shen
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 )
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Sobolev's inequality in central Herz-Morrey-Musielak-Orlicz spaces over metric measure spaces Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2021-01-08 Takao Ohno; Tetsu Shimomura
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
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Complex and quaternionic Cauchy formulas in Koch snowflakes Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2021-01-08 Marisel Avila Alfaro; Abreu Blaya Ricardo
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.
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Homoclinic orbits for first-order Hamiltonian system with local super-quadratic growth condition Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2021-01-06 Wen Zhang; Gang Yang; Fangfang Liao
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.
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Positive solution for a boundary singular semilinear equation with Hardy–Sobolev exponent Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2021-01-06 Ali Rimouche
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.
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Existence of solution for quasilinear Schrödinger equations using a linking structure Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2021-01-06 Edcarlos D. Silva; Jefferson S. Silva
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
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Sliding method for fully nonlinear fractional order equations Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2021-01-05 Yajie Zhang; Feiyao Ma; Weifeng Wo
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.
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Nehari type ground state solutions for periodic Schrödinger–Poisson systems with variable growth Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2020-12-28 Limin Zhang; Xianhua Tang; Sitong Chen
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
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Explicit solutions of a characteristic Goursat problem without the generalized Levi's condition Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2020-12-28 A. Bentrad
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.
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The set of p-harmonic functions in B 1 is total in C k (B̄ 1) Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2020-12-28 J. Villa-Morales
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 ) .
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Half-plane differential-difference elliptic problems with general-kind nonlocal potentials Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2020-12-28 A. B. Muravnik
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
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Anisotropic non-local problems: asymptotic behaviour and existence results Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2020-12-23 Ahlem Yahiaoui; Senoussi Guesmia; Abdelmouhcene Sengouga
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
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Ingham-type theorems for the Dunkl Fourier transforms Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2020-12-21 Mithun Bhowmik; Sanjay Parui; Sanjoy Pusti
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.
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Nonexistence result of nontrivial solutions to the equation −Δu=f(u) Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2020-12-21 Salvador López-Martínez; Alexis Molino
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.
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Schrödinger equations involving fractional p-Laplacian with supercritical exponent Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2020-12-21 Olímpio H. Miyagaki; Sandra I. Moreira; Rônei S. Vieira
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.
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A stability theorem for projective CR manifolds Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2020-12-21 Judith Brinkschulte; C. Denson Hill; Mauro Nacinovich
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
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Koebe and Caratheódory type boundary behavior results for harmonic mappings Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2020-12-16 Daoud Bshouty; Jiaolong Chen; Stavros Evdoridis; Antti Rasila
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
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Mehler kernel approach to Fourier ultra-hyperfunctions Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2020-12-15 Masanori Suwa
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.
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Semilinear fractional-order evolution equations of Sobolev type in the sectorial case Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2020-12-15 Vladimir E. Fedorov; Anna S. Avilovich
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
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Correctness of the definition of the Laplace operator with delta-like potentials Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2020-12-15 B. E. Kanguzhin; K. S. Tulenov
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.
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On the comparison of the Fridman invariant and the squeezing function Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2020-12-15 Feng Rong; Shichao Yang
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
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Global Green's function estimates for the convection–diffusion equation Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2020-12-15 Yurij Alkhutov; Mikhail Surnachev
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.
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Solvability of some integro-differential equations with concentrated sources Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2020-12-15 Vitali Vougalter
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.
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Convolutions of planar harmonic strip mappings Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2020-12-15 Michael Dorff; Samaneh G. Hamidi; Jay M. Jahangiri; Elif Yaar
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.
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Distortion results for a certain subclass of biholomorphic mappings in ℂ n Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2020-12-06 Liangpeng Xiong
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
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Existence of solution for a system involving a singular-nonlocal operator, a singularity and a Radon measure Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2020-12-06 Amita Soni; Sanjoy Datta; K. Saoudi; D. Choudhuri
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
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Fredholm property and essential spectrum of 3-D Dirac operators with regular and singular potentials Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2020-12-06 Vladimir Rabinovich
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
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The Fourier type expansions on tubes Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2020-12-06 Weixiong Mai; Tao Qian
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.
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Multidimensional singular integrals and integral equations in fractional spaces, II Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2020-12-06 N. K. Bliev
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.
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Positive solutions to Schrödinger-Kirchhoff equations with inverse potential Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2020-11-25 Linfeng Luo; Zuji Guo
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
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Existence and multiplicity results for critical anisotropic Kirchhoff-type problems with nonlocal nonlinearities Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2020-11-22 Gelson C. G. dos Santos; Julio R. S. Silva; Suellen Cristina Q. Arruda; Leandro S. Tavares
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
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Stability of the inverse problem for Dini continuous conductivities in the plane Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2020-11-13 Robert McOwen; Bindu K. Veetel
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.
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Overdetermined problems for p-Laplace and generalized Monge–Ampére equations Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2020-11-11 Behrouz Emamizadeh; Yichen Liu; Giovanni Porru
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.
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Moduli inequalities for W 1 n-1,loc-mappings with weighted bounded (q, p)-distortion Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2020-11-10 S. K. Vodopyanov
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.
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Existence and multiplicity results for p(⋅)&q(⋅) fractional Choquard problems with variable order Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2020-11-03 Jiabin Zuo; Alessio Fiscella; Anouar Bahrouni
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 (
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On the stability phenomenon of the Navier-Stokes type equations for elliptic complexes Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2020-11-03 Andrei Parfenov; Alexander Shlapunov
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
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Smoothness of solutions to the mixed problem for elliptic differential-difference equation in cylinder Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2020-11-02 V. V. Liiko; A. L. Skubachevskii
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.
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Regularity of solutions to the Robin problem for differential-difference equations Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2020-11-02 D. A. Neverova
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
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An integral equation method for the Helmholtz problem in the presence of small anisotropic inclusions Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2020-11-02 Houssem Lihiou; Abdessatar Khelifi
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
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A remark on Kirchhoff-type equations in ℝ4 involving critical growth Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2020-10-29 Liu Zeng; Yisheng Huang
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
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On Poleckii-type modular inequalities Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2020-10-29 Mihai Cristea
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
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Solvability of nonlinear problem for some second-order nonstrongly elliptic system Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2020-10-28 Lyailya Zhapsarbayeva; Kordan Ospanov
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
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Nonexistence of nonnegative entire solutions of semilinear elliptic systems Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2020-10-28 Alexander Gladkov; Sergey Sergeenko
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.
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Nonclassical stationary and nonstationary problems with weight Neumann conditions for singular equations with KPZ-nonlinearities Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2020-10-28 A. B. Muravnik
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
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On factorisation of a class of Schrödinger operators Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2020-10-27 Ari Laptev
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.
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Solvability of the Cauchy problem for a pseudohyperbolic system Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2020-10-16 L. N. Bondar; G. V. Demidenko
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.
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On boundary value problems of the Samarskii–Ionkin type for the Laplace operator in a ball Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2020-10-11 Makhmud Sadybekov; Aishabibi Dukenbayeva
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.
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Lyapunov, Hartman-Wintner and De La Vallée Poussin-type inequalities for fractional elliptic boundary value problems Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2020-10-11 Aidyn Kassymov; Mokhtar Kirane; Berikbol T. Torebek
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
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A note on interface formation in singularly perturbed elliptic problems Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2020-10-06 M. Sônego
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.
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On the local and boundary behaviour of mappings of factor spaces Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2020-10-06 Evgeny Sevost'yanov
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
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Infinitely many sign-changing solutions for Choquard equation with doubly critical exponents Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2020-10-02 Senli Liu; Jie Yang; Haibo Chen
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
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Scalar elliptic equations with a singular drift Complex Var. Elliptic Equ. (IF 0.695) Pub Date : 2020-09-29 Misha Chernobai; Timofey Shilkin
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.