Radial Solutions Research Articles

Some results for semi-stable radial solutions of k-Hessian equations with weight on ℝn

We devote this paper to study semi-stable nonconstant radial solutions of $S_k(D^2u)=w(\left \vert x \right \vert )g(u)$ on the Euclidean space $\mathbb {R}^n$. We establish pointwise estimates and necessary conditions for the existence of such solutions (not necessarily bounded) for this equation. For bounded solutions we estimate their asymptotic behaviour at infinity. All the estimates are given in terms of the spatial dimension $n$, the values of $k$ and the behaviour at infinity of the growth rate function of $w$.

On weak solutions to a fractional Hardy–Hénon equation, Part II: Existence

This paper and Hasegawa et al. (2021) treat the existence and nonexistence of stable weak solutions to a fractional Hardy–Hénon equation (−Δ)su=|x|ℓ|u|p−1u in RN, where 0<s<1, ℓ>−2s, p>1, N≥1 and N>2s. In this paper, when p is critical or supercritical in the sense of the Joseph–Lundgren, we prove the existence of a family of positive radial stable solutions, which satisfies the separation property. We also show the multiple existence of the Joseph–Lundgren critical exponent for some ℓ∈(0,∞) and s∈(0,1), and this property does not hold in the case s=1.

A remark on the existence of entire large positive radial solutions to nonlinear differential equations and systems

This paper establishes the existence of entire large positive radial solutions for a class of nonlinear equations and systems. The method employed is based on a successive approximation technique. Our results improve some existing works recently published.

Some weighted fourth-order Hardy-Hénon equations

By using a suitable transform related to Sobolev inequality, we investigate the sharp constants and optimizers in radial space for the following weighted Caffarelli-Kohn-Nirenberg-type inequalities:∫RN|x|α|Δu|2dx≥Srad(N,α)(∫RN|x|−α|u|pα⁎dx)2pα⁎,u∈Cc∞(RN), where N≥3, 4−N<α<2, pα⁎=2(N−α)N−4+α. Then we obtain the explicit form of the unique (up to scaling) radial positive solution Uλ,α to the weighted fourth-order Hardy (for α>0) or Hénon (for α<0) equation:Δ(|x|αΔu)=|x|−αupα⁎−1,u>0inRN. For α≠0, it is known the solutions of above equation are invariant for dilations λN−4+α2u(λx) but not for translations. However we show that if α is a negative even integer, there exist new solutions to the linearized problem, which related to above equation at U1,α, that “replace” the ones due to the translations invariance. This interesting phenomenon was first shown by Gladiali, Grossi and Neves (2013) [26] for the second-order Hénon problem. Finally, as applications, we investigate the remainder term of above inequality and also the existence of solutions to some related perturbed equations.

Parallel schedules for MOC sweep in the neutron transport code PANDAS-MOC

Method of characteristics (MOC) is a commonly applied technique for solving the Boltzmann form of the neutron transport equation. In the PANDAS-MOC neutron transport code, MOC is used to determine the 2D radial solution. However, in the whole-code OpenMP threading hybrid model (WCP) of PANDAS-MOC, it is found that when using the classic parallelism, the MOC sweeping performance is restricted by the overhead incurred by the unbalanced workload and omp atomic clause. This article describes three parallel algorithms for the MOC sweep in the WCP model: the long-track schedule, equal-segment schedule, and no-atomic schedule. All algorithms are accomplished by updating the partition approach and rearranging the sweeping order of the characteristic rays, and their parallel performances are examined by the C5G7 3D core. The results illustrate that the no-atomic schedule can reach 0.686 parallel efficiency when using 36 threads, which is larger than the parallel efficiency obtained in the MPI-only parallelization model.

Radial solutions for p‐Laplacian Neumann problems with gradient dependence

AbstractWe study the radial solutions of the p‐Laplacian Neumann problem with gradient dependence of the type where is either a ball or an annulus, and . By using the topological transversality method together with the barrier strip technique, we obtain existence results of radial solutions to the above problem, where the nonlinearity f does not need to satisfy any growth restriction.

Taxis-driven persistent localization in a degenerate Keller-Segel system

The degenerate Keller-Segel type system is considered in balls with R > 0 and m > 1. Our main results reveal that throughout the entire degeneracy range the interplay between degenerate diffusion and cross-diffusive attraction herein can enforce persistent localization of solutions inside a compact subset of Ω, no matter whether solutions remain bounded or blow up. More precisely, it is shown that for arbitrary and one can find such that if and is nonnegative and radially symmetric with and then a corresponding zero-flux type initial-boundary value problem admits a radial weak solution (u, v), extensible up to a maximal time and satisfying if which has the additional property that In particular, this conclusion is seen to be valid whenever u 0 is radially nonincreasing with

Qualitative analysis on logarithmic Schrödinger equation with general potential

In this paper, we study the existence, uniqueness, nondegeneracy, and some qualitative properties of positive solutions for the logarithmic Schrödinger equations: $$\begin{aligned} -\Delta u+ V(|x|) u=u\log u^2, \quad u\in H^1({\mathbb {R}}^N). \end{aligned}$$ Here $$N\ge 2$$ , and $$V\in C^2((0,+\infty ))$$ is allowed to be singular at 0 and repulsive at infinity (i.e., $$V(r)\rightarrow -\infty $$ as $${r\rightarrow \infty }$$ ). Under some general assumptions, we show the existence, uniqueness, and nondegeneracy of this equation in the radial setting. Specifically, these results apply to singular potentials such as $$V(r)=\alpha _{1}\log r+\alpha _2 r^{\alpha _3}+\alpha _4$$ with $$\alpha _1>1-N$$ , $$\alpha _2, \alpha _3\ge 0$$ , and $$\alpha _4\in {\mathbb {R}}$$ , which is repulsive for $$\alpha _1<0$$ and $$\alpha _2=0$$ . We also investigate the connection between some power-law nonlinear Schrödinger equation with a critical frequency potential and the logarithmic-law Schrödinger equation with $$V(r)=\alpha \log r$$ , $$\alpha >1-N$$ , proving convergence of the unique positive radial solution from the power-type problem to the logarithmic-type problem. Under a further assumption, we also derive the uniqueness and nondegeneracy results in $$H^1({\mathbb {R}}^N)$$ by showing the radial symmetry of solutions.

Solutions to the Dirac equation in Kerr-Newman geometries including the black-hole region

We investigate the Dirac equation in Kerr-Newman space-time, using horizon penetrating coordinates (Eddington-Finkelstein-Coordinates) and the Newman-Penrose formalism to separate the equation into radial and angular systems of ordinary differential equations, and deriving the asymptotics of the radial solutions at infinity and at the Cauchy horizon.

Nodal solutions for an asymptotically linear Kirchhoff-type problem in ℝ N

ABSTRACT In this paper, we are concerned with nodal solutions for a class of Kirchhoff-type problems where , a, b>0, f satisfies some asymptotically linear growth conditions. First of all, when N = 3, b>0 and , b>0 sufficiently small, we attain infinitely many nodal solutions by an equivalent transformation. Based on this, via a new and direct approach, the least energy sign-changing radial solutions can be obtained for the above problem. What's more, we also establish non-existence results of nodal solutions for and b large enough.

An inequality of Bessel functions and applications to transcritical bifurcation problems of nonlinear elliptic equations

We prove a weighted integral inequality of the Bessel function of zero order, which implies the positivity of the integral of the cube of nodal radial eigenfunctions of the Laplacian operator in the planar disk. This solves an open problem in Bonheure et al. (2016) and Corsato et al. (2020). The positivity plays a crucial role in transcritical bifurcation problems of nodal radial solutions of nonlinear elliptic equations.

Normalized solutions to p-Laplacian equations with combined nonlinearities* *Supported by National Natural Science Foundation of China (12031015, 11771428, 12026217, 11871302).

In this paper, we study the p-Laplacian equation with a L p -norm constraint: where N ⩾ 2, a > 0, , , and is a Lagrange multiplier, which appears due to the mass constraint ‖u‖ p = a. We assume that g is odd and L p -supercritical. When and μ > 0, we use Schwarz rearrangement and Ekeland variational principle to prove the existence of positive radial ground states for suitable μ. When and μ > 0 or and μ ⩽ 0, with an additional condition of g, we obtain a positive radial ground state if μ lies in a suitable range, by the Schwarz rearrangement and minimax theorems. Via a fountain theorem type argument, with suitable , we obtain infinitely many radial solutions for any N ⩾ 2 and establish the existence of infinitely many nonradial sign-changing solutions for N = 4 or N ⩾ 6. In any case mentioned above, the range of μ depends on the value of a: |μ| can be large if a > 0 is small.

Quasilinear Schrödinger equations with bounded coefficients

We study quasilinear elliptic equations of the form , where is a bounded function, V(x) is allowed to be singular at the origin, and is a general nonlinearity. Such type of equations has been derived as models of several physical phenomena corresponding to various types of . Despite the lack of a priori compactness condition for the energy functional, we develop a new variational approach to deal with the issues of multiple radial solutions, nonradial solutions and sign-changing solutions.

Existence of infinitely many positive radial solutions for an iterative system of nonlinear elliptic equations on an exterior domain

Existence of infinitely many positive radial solutions for an iterative system of nonlinear elliptic equations on an exterior domain

Existence of radial weak solutions to Steklov problem involving Leray–Lions type operator

We make use of variational methods to prove the existence of at least one positive radial increasing weak solution to a Leray–Lions type problem under Steklov boundary conditions.

Positive radial solutions for Dirichlet problems via a Harnack‐type inequality

We deal with the existence and localization of positive radial solutions for Dirichlet problems involving ‐Laplacian operators in a ball. In particular, ‐Laplacian and Minkowski‐curvature equations are considered. Our approach relies on fixed point index techniques, which work thanks to a Harnack‐type inequality in terms of a seminorm. As a consequence of the localization result, it is also derived the existence of several (even infinitely many) positive solutions.

Characterising the orbit and circumstellar environment of the high-mass binary MWC 166 A

Context. Stellar evolution models are highly dependent on accurate mass estimates, especially for highly massive stars in the early stages of stellar evolution. The most direct method for obtaining model-independent stellar masses is derivation from the orbit of close binaries. Aims. Our aim was to derive the first astrometric plus radial velocity orbit solution for the single-lined spectroscopic binary star MWC 166 A, based on near-infrared interferometry over multiple epochs and ∼100 archival radial velocity measurements, and to derive fundamental stellar parameters from this orbit. A supplementary aim was to model the circumstellar activity in the system from K band spectral lines. Methods. The data used include interferometric observations from the VLTI instruments GRAVITY and PIONIER, as well as the MIRC-X instrument at the CHARA Array. We geometrically modelled the dust continuum to derive relative astrometry at 13 epochs, determine the orbital elements, and constrain individual stellar parameters at five different age estimates. We used the continuum models as a base to examine differential phases, visibilities, and closure phases over the Br γ and He I emission lines in order to characterise the nature of the circumstellar emission. Results. Our orbit solution suggests a period of P = 367.7 ± 0.1 d, approximately twice as long as found with previous radial velocity orbit fits. We derive a semi-major axis of 2.61 ± 0.04 au at d = 990 ± 50 pc, an eccentricity of 0.498 ± 0.001, and an orbital inclination of 53.6 ± 0.3°. This allowed the component masses to be constrained to M1 = 12.2 ± 2.2 M⊙ and M2 = 4.9 ± 0.5 M⊙. The line-emitting gas was found to be localised around the primary and is spatially resolved on scales of ∼11 stellar radii, where the spatial displacement between the line wings is consistent with a rotating disc. Conclusions. The large spatial extent and stable rotation axis orientation measured for the Br γ and He I line emission are inconsistent with an origin in magnetospheric accretion or boundary-layer accretion, but indicate an ionised inner gas disc around this Herbig Be star. We observe line variability that could be explained either with generic line variability in a Herbig star disc or V/R variations in a decretion disc scenario. We have also constrained the age of the system, with relative flux ratios suggesting an age of ∼(7 ± 2)×105 yr, consistent with the system being composed of a main-sequence primary and a secondary still contracting towards the main-sequence stage.

Asymptotic behavior of blowing-up radial solutions for quasilinear elliptic systems arising in the study of viscous, heat conducting fluids

In this paper, we deal with the following quasilinear elliptic system involving gradient terms in the form: $$ \begin{cases} \Delta_p u= v^m| \nabla u |^\alpha & \text{in}\quad \Omega\\ \Delta_p v= v^\beta| \nabla u |^q & \text{in}\quad \Omega, \end{cases} $$ where $\Omega\subset\mathbb{R}^N\ (N\geq 1)$ is either equal to $ \mathbb{R}^N $ or equal to a ball $B_R$ centered at the origin and having radius $R>0$, $1 < p < \infty$, $m,q > 0$, $\alpha\geq 0$, $0\leq \beta\leq m$ and $\delta:=(p-1-\alpha)(p-1-\beta)-qm \neq 0$. Our aim is to establish the asymptotics of the blowing-up radial solutions to the above system. Precisely, we provide the accurate asymptotic behavior at the boundary for such blowing-up radial solutions. For that, we prove a strong maximal principle for the problem of independent interest and study an auxiliary asymptotically autonomous system in $\mathbb{R}^3$.

Ground state solutions for critical Schrödinger equations with Hardy potential

In this paper, we investigate the following Schrödinger equation 0.1 −Δu−μ|x|2u=g(u)+|u|2*−2uinRN\\{0}, where N ⩾ 3, , is called the critical Sobolev exponent and g satisfies some appropriate subcritical conditions. For any , we prove that problem (0.1) has a positive radial ground state solution, which possesses exponential decaying property at infinity and blow-up property at origin. Moreover, for any sequence {μ n } ⊂ (0, +∞) satisfying μ n → 0+, the sequence of ground state solutions to problem (0.1) converges to a ground state solution of 0.2 −Δu=g(u)+|u|2*−2uinRN. when μ < 0, we prove that the mountain pass level of problem (0.1) in cannot be achieved. Further, we obtain a ground state radial solution of problem (0.1) whose energy is strictly greater than the mountain pass level in . Also, for any sequence {μ n } ⊂ (0, +∞) satisfying μ n → 0+, the sequence of ground state radial solutions to problem (0.1) converges to a ground state radial solution of the limiting problem as n → ∞.

Remarks on Radial Solutions of a Parabolic Gelfand-Type Equation

We consider an equation with exponential nonlinearity under the Dirichlet boundary condition. For a one- or two-dimensional domain, a global solution has been obtained. In this paper, to extend the result to a higher dimensional case, we concentrate on the radial solutions in an annulus. First, we construct a time-local solution with an abstract theory of differential equations. Next, we show that decreasing energy exists in this problem. Finally, we establish a global solution for the sufficiently small initial value and parameter by Sobolev embedding and Poincaré inequalities together with some technical estimates. Moreover, when we take the smaller parameter, we prove that the global solution tends to zero as time goes to infinity.