Showing papers in "Complex Variables and Elliptic Equations in 2021"
TL;DR: In this article, the existence/multiplicity results for the variable order nonlocal Choquard problem with variable exponents were studied and the existence and multiplicity results were derived.
Abstract: 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,...
29 citations
TL;DR: In this article, the polarization technique was used to prove modular and norm Polya-Szego inequalities in general fractional Orlicz-Sobolev spaces.
Abstract: In this article, we prove modular and norm Polya–Szego inequalities in general fractional Orlicz–Sobolev spaces by using the polarization technique. We introduce a general framework which includes ...
26 citations
TL;DR: Li and Barlund as discussed by the authors introduced a distance function which is a metric for triangle inequality, which they used to answer a question about triangle inequality suggested by R. Li and R. Barlund.
Abstract: 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...
17 citations
TL;DR: In this paper, the existence of non-trivial solutions for a class of p-biharmonic problems is proved by using a variational approach combined with the Nehari manifold method and fibreing maps.
Abstract: 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 ...
15 citations
TL;DR: The quaternionic offset linear canonical transform (QOLCT) as mentioned in this paper can be defined as a generalization of the QOLCT and is derived from the Quaternionic Linear Canonical Transform (QLCT).
Abstract: The quaternionic offset linear canonical transform (QOLCT) can be defined as a generalization of the quaternionic linear canonical transform (QLCT). In this paper, we define the QOLCT, we derive th...
14 citations
TL;DR: In this article, the following Chern-Simons-Schrodinger system is investigated: Δu+u+λ∫|x|∞h(s)su2(s),ds+h2(|x |)|x|2u=f(x,u)inR2, where λ>0, h(s)=∫0st2u2(t)dt and the nonlinearity f(x,s)∈C(R2×R,R) beha...
Abstract: In this paper, we investigate the following Chern–Simons–Schrodinger system −Δu+u+λ∫|x|∞h(s)su2(s)ds+h2(|x|)|x|2u=f(x,u)inR2, where λ>0, h(s)=∫0st2u2(t)dt and the nonlinearity f(x,s)∈C(R2×R,R) beha...
13 citations
TL;DR: In this paper, the triangular ratio metric is studied in a domain G⊊Rn, n≥2, and several sharp bounds are proven for this metric, especially in the case where the domain is the unit disk of the complex plane.
Abstract: The triangular ratio metric is studied in a domain G⊊Rn, 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 res...
13 citations
12 citations
TL;DR: In this paper, a class of fractional Laplacian problems of the form (−Δ)p1(x,.)su+−Δp2(x+γ(x)+λf(x + γ(x),u) in Ω⊂RN, n ≥ 2, where n is a bounded domain and γ is a constant.
Abstract: 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 ...
11 citations
TL;DR: In this paper, the Henon-Hardy type system on Rn was studied, and Liouville theory was proved for Rn type systems with n ≥ 2, n>α, 0 < α≤2 or α = 2m.
Abstract: In this paper, we are concerned with the Henon–Hardy type systems on Rn: (−Δ)α2u(x)=|x|avp(x),u(x)≥0, x∈Rn,(−Δ)α2v(x)=|x|buq(x),v(x)≥0, x∈Rn, where n≥2, n>α, 0<α≤2 or α=2m. We prove Liouville theor...
11 citations
TL;DR: In this paper, a fractional wave equation for hypoelliptic operators with a singular mass term depending on the spacial variable is considered, and it is shown that it has a very weak solution.
Abstract: In this paper we consider a fractional wave equation for hypoelliptic operators with a singular mass term depending on the spacial variable and prove that it has a very weak solution. Such analysis can be conveniently realised in the setting of graded Lie groups. The uniqueness of the very weak solution, and the consistency with the classical solution are also proved, under suitable considerations. This extends and improves the results obtained in the first part of this work which was devoted to the classical Euclidean Klein-Gordon equation.
TL;DR: 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 in this article.
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 line...
TL;DR: In this article, it was shown 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.
Abstract: 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 condi...
TL;DR: In this paper, the existence of a weak solution for nonlinear p-sub-Laplacian equations with Dirichlet boundary condition on the Heisenberg group was proved.
Abstract: In this note, we prove the existence of a weak solution for nonlinear p-sub-Laplacian equations with the Dirichlet boundary condition on the Heisenberg group. In the proof, we use a version of the ...
TL;DR: In this article, the L q -mixed problem in domains in R n with C 1, 1 -boundary is considered and the boundary between the sets where we specify Neumann and Dirichlet data is Lipschitz.
Abstract: We consider the L q -mixed problem in domains in R n with C 1 , 1 -boundary. We assume that the boundary between the sets where we specify Neumann and Dirichlet data is Lipschitz. With these assump...
TL;DR: The n-dimensional generalized Hartogs triangles are domains defined by Hpn:={(z1,…,zn)∈Cn:|z1|p1<⋯<|zn|pn<1}.
Abstract: 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 Berg...
TL;DR: In this paper, the authors consider an elliptic equation driven by a p-Laplacian-like operator on an n-dimensional Riemannian manifold and show that the growth condition on the right-hand side depends on the geometry of the manifold.
Abstract: We consider an elliptic equation driven by a p-Laplacian-like operator, on an n-dimensional Riemannian manifold. The growth condition on the right-hand side of the equation depends on the geometry ...
TL;DR: In this paper, the authors consider general multivector elements of Clifford algebras Cl(3,0), Cl(1,2) and Cl(0,3), and look for possibilities to factorize multivectors into products of blades, idempotents and exponentials, where the exponents are frequently blades of grades zero to n.
Abstract: In this paper we consider general multivector elements of Clifford algebras Cl(3,0), Cl(1,2) and Cl(0,3), and look for possibilities to factorize multivectors into products of blades, idempotents and exponentials, where the exponents are frequently blades of grades zero (scalar) to n (pseudoscalar).
TL;DR: In this article, the existence of anisotropic nonlocal solutions to the nonlocal problem is studied. But the authors do not consider the case where Ω is a smooth smooth polygon.
Abstract: 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...
TL;DR: In this paper, the existence of solutions for the following critical fractional pq(−Δ)ps1v+Δ+1v + δ + ε+δ q(− Δ)ps 1v + − ε q(δ)q(n) + − Δ q(n−ε) q( − δ)pq( − Δ)qs 1v+ ε)qs + −Δ q (n− ε)-q
Abstract: In this article, we are concerned with the existence of solutions for the following critical fractional pq(−Δ)ps1v+(−Δ)qs...
TL;DR: In this article, it was shown that ζ is not a solution of any non-trivial algebraic differential equation whose coefficients are polynomials in C, l>n>α≥0 a.
Abstract: In this paper, we prove that ζ is not a solution of any non-trivial algebraic differential equation whose coefficients are polynomials in Γ(α),Γ(n),Γ(l) over the ring of polynomials in C, l>n>α≥0 a...
TL;DR: In this article, a singular problem involving the p$(x)$-Laplace operator is studied, where the p(x)-Laplace operators is a nonconstant continuous function.
Abstract: We study the following singular problem involving the p$(x)$-Laplace operator $\Delta_{p(x)}u= div(|
abla u|^{p(x)-2}
abla u)$, where $p(x)$ is a nonconstant continuous function, \begin{equation}
onumber {{(\rm P_\lambda)}} \left\{\begin{aligned} - \Delta_{p(x)} u & = a(x)|u|^{q(x)-2}u(x)+ \frac{\lambda b(x)}{u^{\delta(x)}} \quad\mbox{in}\,\Omega,\\ u &>0 \quad\mbox{in}\,\Omega, \\ u & =0 \quad\mbox{on}\,\partial\Omega.\end{aligned} \right. \end{equation} Here, $\Omega$ is a bounded domain in $\mathbb{R}^{N\geq2}$ with $C^2$-boundary, $\lambda$ is a positive parameter, $a(x), b(x) \in C(\overline{\Omega})$ are positive weight functions with compact support in $\Omega$, and $\delta(x),$ $p(x),$ $q(x) \in C(\overline{\Omega})$ satisfy certain hypotheses ($A_{0}$) and ($A_{1}$). We apply the Nehari manifold approach and some new techniques to establish the multiplicity of positive solutions for problem ${{(\rm P_\lambda)}}$.
TL;DR: In this article, an Almansi-type decomposition for polynomials with Clifford coefficients was presented for slice-regular functions on Clifford algebras, and more generally for slice regular functions on the Clifford algebra.
Abstract: 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, pub...
TL;DR: In this article, the authors proved a positive solution to a questio... problem for the two-index scale of weighted bounded (q, p ) -distortion under minimal regularity.
Abstract: 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 questio...
TL;DR: In this paper, 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 equation is defined as follows:
Abstract: 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 equation...
TL;DR: The authors generalize some earlier extensions of the modular inequality of Poleckii, which is a basic tool in the geometric theory of mappings and was used first in the well-known theory of quasiregular mappings.
Abstract: 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,...
TL;DR: Wei-Wu et al. as discussed by the authors studied the Choquard equation with a local perturbation and proved nonexistence, existence and symmetry of normalized ground states, by using the mountain pass lemma, the Pohožaev constraint method, the Schwartz symmetrization rearrangements and some theories of polarizations.
Abstract: We study the Choquard equation with a local perturbation \begin{equation*} -\Delta u=\lambda u+(I_\alpha\ast|u|^p)|u|^{p-2}u+\mu|u|^{q-2}u,\ x\in \mathbb{R}^{N} \end{equation*} having prescribed mass \begin{equation*} \int_{\mathbb{R}^N}|u|^2dx=a^2. \end{equation*} For a $L^2$-critical or $L^2$-supercritical perturbation $\mu|u|^{q-2}u$, we prove nonexistence, existence and symmetry of normalized ground states, by using the mountain pass lemma, the Pohožaev constraint method, the Schwartz symmetrization rearrangements and some theories of polarizations. In particular, our results cover the Hardy-Littlewood-Sobolev upper critical exponent case $p=(N+\alpha)/(N-2)$. Our results are a nonlocal counterpart of the results in \cite{{Li 2021-4},{Soave JFA},{Wei-Wu 2021}}.
TL;DR: In this paper, the extension of the Schwarz representation formula to simply connected domains with harmonic Green function and its polyanalytic generalization is not valid in general, and they do hold only for certa...
Abstract: The extension of the Schwarz representation formula to simply connected domains with harmonic Green function and its polyanalytic generalization is not valid in general. They do hold only for certa...
TL;DR: In this article, the sharp Bohr-Rogosinski inequality, improved Bohr inequality, refined Bohr inequalities and Bohr type inequality were obtained for the class of complex-valued harmonic mappings.
Abstract: Let $ \mathcal{H} $ be the class of complex-valued harmonic mappings $ f=h+\bar{g}$ defined in the unit disk $ \mathbb{D} : =\{z\in\mathbb{C} : |z| -M+|zg^{\prime\prime}(z)|,\; z \in \mathbb{D}\; \mbox{and}\;\; M>0\}.
$$ In this paper, we obtain the sharp Bohr-Rogosinski inequality, improved Bohr inequality, refined Bohr inequality and Bohr-type inequality for the class $ \mathcal{P}_{\mathcal{H}}^{0}(M) $.