Radial Solutions Research Articles

Asymptotic formula for the Morse index of radial solutions on expanding annuli: Allen-Cahn equation and scalar filed equation

Let AR⊂RN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_R\\subset \\mathbb {R}^N$$\\end{document}, N≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N\\ge 2$$\\end{document}, be an annulus with inner radius R and outer radius R+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R+1$$\\end{document}. We are concerned with the elliptic Neumann problem ε2Δu+f(u)=0forx∈AR,∂u∂n=0forx∈∂AR,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} \\varepsilon ^2\\Delta u+f(u)=0 & \ ext {for}\\ x\\in A_R,\\\\ \\frac{\\partial u}{\\partial n}=0 & \ ext {for}\\ x\\in \\partial A_R, \\end{array}\\right. } \\end{aligned}$$\\end{document}where ε>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon >0$$\\end{document} is a small constant. In particular, the Allen-Cahn equation f(u)=u-u3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(u)=u-u^3$$\\end{document} and the scalar field equation f(u)=-u+up\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(u)=-u+u^p$$\\end{document}, p>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p>1$$\\end{document}, are studied. We establish sharp asymptotic formulas of the Morse index of n-mode radial solutions as R→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R\\rightarrow \\infty $$\\end{document}. In the case of the scalar field equation the first n eigenvalues of the linearization around n-mode solutions of the one-dimensional problem ε2u′′-u+up=0for0<x<1,u′(0)=u′(1)=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} \\varepsilon ^2u''-u+u^p=0 & \ ext {for}\\ 0<x<1,\\\\ u'(0)=u'(1)=0 \\end{array}\\right. } \\end{aligned}$$\\end{document}become important. We show that, as ε→0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon \\rightarrow 0$$\\end{document}, the first ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell $$\\end{document} eigenvalues converge to -(p+3)(p-1)/4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$-(p+3)(p-1)/4$$\\end{document} and the other n-ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n-\\ell $$\\end{document} eigenvalues converge to 0, where ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell $$\\end{document} is the number of the local maximum points of an n-mode solution u(x) on [0, 1].

Dynamics of massive and massless particles in the spacetime of a wiggly cosmic dislocation

In this paper, we investigate the spacetime containing both small-scale structures (wiggles) and spatial dislocation, forming a wiggly cosmic dislocation. We study the combined effects of these features on the dynamics of massive and massless particles. Our results show that while wiggles alone lead to bound states and dislocation introduces angular momentum corrections, their coupling produces more complex effects, influencing both particle motion and wave propagation. Notably, this coupling significantly modifies radial solutions and eigenvalues, with the direction of motion or propagation becoming a critical factor in determining the outcomes. Numerical solutions reveal detailed aspects of particle dynamics as functions of dislocation and string parameters, including plots of trajectories, probability densities, and energy levels. These findings deepen our understanding of how a wiggly cosmic dislocation shapes particle dynamics, suggesting new directions for theoretical exploration in cosmological models.

Differential Operators on Non-compact Harmonic Manifolds

We study the algebra of differential operators on non-compact simply connected harmonic manifolds and provide sufficient conditions for them to have a radial fundamental solution and be surjective on the space of smooth function. Furthermore, we show that the algebra of differential operators that commute with taking averages over geodesic spheres is generated by the Laplacian. As an application of this, we show that the heat-semi group is dense in the radial L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1$$\\end{document} space of a non-compact simply connected harmonic manifold. In the process, we provide a characterisation of the eigenfunctions of the Laplacian and therefore of the eigenfunctions of all differential operators commuting with taking spherical averages.

On type I blowup of some nonlinear heat equations with a potential

In this paper, we are concerned with the following initial-boundary value problem{ut=Δu+Q(|x|)|u|p−1u,x∈BR,t>0u(x,t)=0,x∈∂BR,t>0u(x,0)=u0(x),x∈BR, where p≥ps:=N+2N−2, u0∈L∞(BR), and Q(r)∈C1([0,R]), 0<C_≤Q(r)≤C‾<∞,Q′(r)≤0. We extend the asymptotic behavior results, which is well-known when Q is constant according to Matano-Merle (cf. [25]), for the blow-up solutions. More precisely, we show that when ps≤p<p⁎, the blowup of radial solution to this problem is always of Type I. This result partially generalizes the conclusions in [25] for Q≡1. This extension is nontrivial due to the appearance of Q. The quasi-monotonicity formula established by the third author and Cheng in [8] allows us to use an energy method to get a priori estimates on the rescaled solutions. The contraction mapping principle shows the existence of singular stationary solutions to an associated elliptic equation with a potential. In the end, the properties of zero number for solutions lead to the nonexistence of type II singularity for the problem.

Nonexistence of radial positive solutions for a class of nonpositone problems in a ball

Nonexistence of radial positive solutions for a class of nonpositone problems in a ball

Power distribution system restoration based on soft open points and islanding by distributed generations

Abstract A powerful and efficient program for restoring the electrical distribution system by effectively utilizing the maximum capabilities within the system, including soft open points, distributed generation resources, and network configuration changes, can significantly reduce both the quantity and duration of lost loads caused by permanent faults. In this research study, we address the issue of distribution network restoration with a focus on soft open points (SOPs) and intentional islanding using distributed generations. This problem is formulated as a constrained optimization problem in order to determine the optimal distribution network configuration, quality of intentional islanding through DGs, control function of SOPs, and amount of load shedding. The objective is to minimize lost load while minimizing switching operations and preferably minimal islanding. Given that this problem involves multiple complexities such as combinatorial nature, mixed-integer variables, non-linearity, non-convexity along with numerous variables and constraints; we employ an evolutionary method known as simulated annealing (SA) algorithm to solve it without any simplifications or assumptions about convexity. To enhance efficiency during implementation of SA algorithm, Kruskal’s algorithm is utilized for generating radial solutions which restricts search space to feasible solutions resulting in quicker attainment of high-quality optimal solutions. Finally, the outcomes of implementing the suggested approach on the 69-bus IEEE distribution system are presented and examined. It is demonstrated that by leveraging the potential of soft open points in load transfer control and network voltage regulation, along with utilizing intentional islanding capability provided by distributed generation resources, the restoration capability of the distribution system can be greatly enhanced.

Triple positive radial solutions arising from biharmonic elliptic systems

Triple positive radial solutions arising from biharmonic elliptic systems

Some remarks about the Morse index for convex Hamiltonian systems

We investigate the (linearized) Morse index of solutions to Hamiltonian systems, with a focus on convex Hamiltonian functions and sign-changing radial solutions. For strongly coupled systems, we describe the profile of the radial solutions and give an estimate of their Morse index.

Existence and uniqueness of positive solution to a new class of nonlocal elliptic problem with parameter dependency

In this paper, we prove that under weak assumptions on the reaction terms and diffusion coefficients, a positive solution exists for a one-dimensional case and a positive radial solution to a multidimensional case of a nonlocal elliptic problem. Additionally, we establish the uniqueness of the solution, with the fixed point theorem being the primary tool employed. Our results are new and generalize several existing results.

Radial Solutions for p-k-Hessian Equations and Systems with Gradient Term

This paper studies the existence of entire radial solutions to the p-k-Hessian equation with nonlinear gradient term σk(λDi|Du|p-2Dj(u)+α|∇u|(p-1)k=a(|x|)fk(u),x∈Rn,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\sigma _{k}\\left. (\\lambda \\left( D_{i}\\left( |D u|^{p-2} D_{j}(u)\\right) \\right) +\\alpha |\ abla u|^{(p-1) k}\\right. =a(|x|) f^{k}(u), ~~x \\in \\mathbb {R}^{n}, \\end{aligned}$$\\end{document}and system with nonlinear gradient term σk(λDi|Du|p-2Dj(u)+α|∇u|(p-1)k=a(|x|)fk(v),x∈Rn,σk(λDi|Dv|p-2Dj(v)+β|∇v|(p-1)k=b(|x|)gk(u),x∈Rn.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{l} \\sigma _{k}\\left. (\\lambda \\left( D_{i}\\left( |D u|^{p-2} D_{j}(u)\\right) \\right) +\\alpha |\ abla u|^{(p-1) k}\\right. =a(|x|) f^{k}(v), ~~x \\in \\mathbb {R}^{n}, \\\\ \\sigma _{k}\\left. (\\lambda \\left( D_{i}\\left( |D v|^{p-2} D_{j}(v)\\right) \\right) +\\beta |\ abla v|^{(p-1) k}\\right. =b(|x|) g^{k}(u), ~~x \\in \\mathbb {R}^{n}. \\end{array}\\right. \\end{aligned}$$\\end{document}By adopting monotone iteration method, we derive the existence and asymptotic behavior of the radial solutions.

Nontrivial solutions for multiparameter k-Hessian systems

This paper is devoted to studying the coupled system of multiparameter k-Hessian equations where B = {x ∈ ℝ n : |x| < 1}, λ1, λ2 are positive parameters, ρi (i = 1, 2) is singular near the boundary ∂B, f i(i = 1, 2) satisfies a combined superlinear condition at ∞. Employing the fixed point theorem of Krasnosel’skii type in Banach space, the existence and multiplicity of nontrivial radial solutions are established. In particular, the dependence of the solutions on the parameters λ1, λ2 is also discussed. Finally, we study the nonexistence results of nontrivial radial solutions for the case λ1, λ2 ≫ 1.

The exact relativistic scalar quasibound states of the dyonic Kerr–Sen black hole: quantized energy, and Hawking radiation

We consider Klein–Gordon equation in the Dyonic Kerr–Sen black hole background, which is the charged rotating axially symmetric solution of the Einstein–Maxwell–Dilaton–Axion theory of gravity. The black hole incorporates electric, magnetic, dilatonic and axionic charges and is constructed in 3+1 dimensional spacetime. We begin our investigations with the construction of the scalar field’s governing equation, i.e., the covariant Klein–Gordon equation. With the help of the ansatz of separation of variables, we successfully separate the polar part, and find the exact solution in terms of Spheroidal Harmonics, while the radial exact solution is obtained in terms of the Confluent Heun function. The quantization of the quasibound state is done by applying the polynomial condition of the Confluent Heun function that gives rise to discrete complex-valued energy levels for massive scalar fields. The real part is the scalar field relativistic quantized energy, while the imaginary part represents the quasibound states’s decay. We present all of the sixteen possible exact energy solutions for both massive and massless scalars. We also present the investigation the Hawking radiation of the Dyonic Kerr–Sen black hole’s apparent horizon, via the Sigurd–Sannan method by making use of the obtained exact scalar wave functions. The radiation distribution function, and the Hawking temperature are also obtained.

Existence and asymptotic behavior of nontrivial p-k-convex radial solutions for p-k-Hessian equations

Existence and asymptotic behavior of nontrivial p-k-convex radial solutions for p-k-Hessian equations

The Schrödinger–Born–Infeld system: Attractive case

In this paper we consider a system describing a quantum particle self interacting with the Born Infeld electromagnetic field. The existence of a radial ground state solution is proved in the attractive case.

Radial solutions for Neumann problems involving prescribed mean curvature operator in a ball and in an annular domain

Using topological transversality method together with barrier strip technique and cut-off technique, we obtain new existence and uniqueness results of radial solutions to the Neumann problems involving prescribed mean curvature operator div∇v1+|∇v|2=f|x|,v,dvdrinΩ,∂v∂n=0on∂Ω,where Ω={x∈RN:R1<|x|<R2}(0≤R1<R2,N≥2), f:[R1,R2]×R2→R is continuous. Meanwhile, we demonstrate the importance of our results through an illustrative example.

On p-k-convex solutions for the p-k-Hessian system with the gradient term

In this paper, we consider a class of nonlinear p-k-Hessian systems with the gradient term where γ > 0, p > 1, and 1 ≤ k ≤ N is an integer. We show some existence, boundedness and blow-up results of the entire positive p-k-convex radial solutions based on the monotone iterative method. Finally, we present an example to illustrate our result.

Relativistic scalar fields canonical quantization in Einstein–Yang–Mills–Higgs’s rotating black hole space-time

The quantum theory of relativistic mechanics to deal with the scalar fields behavior in a curved space-time is represented by the Klein–Gordon equation. In this work, we will investigate the gravitationally bound states of massive and massless scalar fields around a Einstein–Yang–Mills–Higgs’s rotating black hole. After applying the standard separation of variables ansatz, we will show in detail how to obtain the novel exact solutions of the radial part of the governing Klein–Gordon equation and express the radial solution in terms of the Confluent Heun functions. By applying the bound state boundary conditions, the Confluent Heun functions are reduced to be polynomials that lead to energy quantization. We find that the scalar fields have complex-valued energy levels that indicate the decaying/growing bound states known as quasibound states. In the end, using the exact radial solution, we derive the radiation distribution function of the black hole’s outer horizon to obtain the equation of the Hawking temperature.

Scalar's quasibound states in cosmological black hole background

In this letter, we present in detail a novel exact solution of scalar quasibound states of a static spherically symmetric Schwarzschild de Sitter black hole. We investigate and work out the governing covariant scalar field wave equation, i.e., the Klein-Gordon equation and isolate the radial equation. In this letter, we show in detail, the derivation of the successfully obtained exact radial wave solutions which are expressed in terms of General Heun functions. Having the exact solutions in hand, the energy levels expression is obtained from special function's polynomial condition. In the last part, the Hawking radiation is investigated via the Damour-Ruffini method and the Hawking temperature is obtained from radiation distribution function.

Topological defects on solutions of the non-relativistic equation for extended double ring-shaped potential

In this study, we focus into the non-relativistic wave equation described by the Schrödinger equation, specifically considering angular-dependent potentials within the context of a topological defect background generated by a cosmic string. Our primary goal is to explore quasi-exactly solvable problems by introducing an extended ring-shaped potential. We utilize the Bethe ansatz method to determine the angular solutions, while the radial solutions are obtained using special functions. Our findings demonstrate that the eigenvalue solutions of quantum particles are intricately influenced by the presence of the topological defect of the cosmic string, resulting in significant modifications compared to those in a flat space background. The existence of the topological defect induces alterations in the energy spectra, disrupting degeneracy. Afterwards, we extend our analysis to study the same problem in the presence of a ring-shaped potential against the background of another topological defect geometry known as a point-like global monopole. Following a similar procedure, we obtain the eigenvalue solutions and analyze the results. Remarkably, we observe that the presence of a global monopole leads to a decrease in the energy levels compared to the flat space results. In both cases, we conduct a thorough numerical analysis to validate our findings.

Notes on asymptotic behavior of radial solutions for some weighted elliptic equations on the annulus

Notes on asymptotic behavior of radial solutions for some weighted elliptic equations on the annulus