Method Of Moving Planes Research Articles

On the Classification of Entire Solutions to the Critical Lane–Emden Equation

The positive solutions to the critical Lane–Emden equation in Rn have been classified completely by the moving plane method. Afterwards, with additional conditions such as limited energy or volume, there is proof that relies entirely on integration by parts and inequalities techniques. In this paper, the authors provide a new approach to obtain the same classification results without further assumptions.

Monotone Positive Radial Solution of Double Index Logarithm Parabolic Equations

This article mainly studies the double index logarithmic nonlinear fractional g-Laplacian parabolic equations with the Marchaud fractional time derivatives ∂tα. Compared with the classical direct moving plane method, in order to overcome the challenges posed by the double non-locality of space-time and the nonlinearity of the fractional g-Laplacian, we establish the unbounded narrow domain principle, which provides a starting point for the moving plane method. Meanwhile, for the purpose of eliminating the assumptions of boundedness on the solutions, the averaging effects of a non-local operator are established; then, these averaging effects are applied twice to ensure that the plane can be continuously moved toward infinity. Based on the above, the monotonicity of a positive solution for the above fractional g-Laplacian parabolic equations is studied.

On a class of Kirchhoff type logarithmic Schrödinger equations involving the critical or supercritical Sobolev exponent

In this paper, we study a class of Kirchhoff type logarithmic Schrödinger equations involving the critical or supercritical Sobolev exponent. Such problems cannot be studied by applying variational methods in a standard way, because the nonlinearities do not satisfy the Ambrosetti-Rabinowitz condition and change sign. Moreover, the appearance of the logarithmic term makes the associated energy functional lose differentiable in the sense of Gateaux. By analyzing the structure of the Nehari manifold and developing some analysis techniques, the above obstacles are overcome in subtle ways and several existence result are obtained. Furthermore, we investigate the regularity, the monotonicity, and the symmetric properties of the solutions via the iterative technique and the moving plane method.

Moving plane method for varifolds and applications

abstract: In this paper, we introduce a version of the moving plane method that applies to potentially quite singular hypersurfaces, generalizing the classical moving plane method for smooth hypersurfaces. Loosely speaking, our version for varifolds shows that smoothness and symmetry at infinity (respectively at the boundary) can be promoted to smoothness and symmetry in the interior. The key feature, in contrast with the classical formulation of the moving plane principle, is that smoothness is a conclusion rather than an assumption. We implement our moving plane method in the setting of compactly supported varifolds with smooth boundary and in the setting of varifolds without boundary. A key ingredient is a Hopf lemma for stationary varifolds and varifolds of constant mean curvature. Our Hopf lemma provides a new tool to establish smoothness of varifolds, and works in arbitrary dimensions and without any stability assumptions. As applications of our new moving plane method, we prove varifold uniqueness results for the catenoid, spherical caps, and Delaunay surfaces that are inspired by classical uniqueness results by Schoen, Alexandrov, Meeks and Korevaar-Kusner-Solomon. We also prove a varifold version of Alexandrov's Theorem for compactly supported varifolds of constant mean curvature in hyperbolic space.

Ground state solutions for Choquard equations with Bessel potential

Firstly, we study the existence, regularity and decay of positive solutions for the elliptic system { − Δ u + u = vg ( u ) in R N , − Δv + v = G ( u ) in R N , where N ≥ 3 , g satisfies Berestycki-Lions type conditions. In Sobolev space, the system can be transformed into the Choquard equations with Bessel potential − Δ u + u = ( g 2 ⋆ G ( u ) ) g ( u ) in R N , where g 2 is the fundamental solution of the linear operator − Δ + I on R N . Moreover, by assuming that g is nondecreasing, we obtain the radial symmetry and monotonic property of positive solutions by moving plane method, and the existence of ground state solutions by non-Nehari manifold method.

Radial solvability for Pucci-Lane-Emden systems in annuli

We establish a priori bounds, existence and qualitative behaviour of positive radial solutions in annuli for a class of nonlinear systems driven by Pucci extremal operators and Lane-Emden coupling in the superlinear regime. Our approach is purely nonvariational. It is based on the shooting method, energy functionals, spectral properties, and on a suitable criteria for locating critical points in annular domains through the moving planes method that we also prove.

Rigidity for measurable sets

Let Ω⊂Rd, d≥2, be a set with finite Lebesgue measure such that, for a fixed radius r>0, the Lebesgue measure of Ω∩Br(x) is equal to a positive constant when x varies in the essential boundary of Ω. We prove that Ω is a ball (or a finite union of equal balls) provided it satisfies a nondegeneracy condition, which holds in particular for any set of diameter larger than r which is either open and connected, or of finite perimeter and indecomposable. The proof requires reinventing each step of the moving planes method by Alexandrov in the framework of measurable sets.

Symmetry results of solutions for elliptic systems with linear couplings

This paper is devoted to study the symmetry and monotonicity of positive solutions for linear coupling elliptic systems in a ball in R N . Using the Alexandrov–Serrin method of moving planes combined with the strong maximum principle, we prove that the solutions of elliptic systems with linear couplings in a ball are symmetric w.r.t. 0 and radially decreasing. For our problems, the tangential gradient of solutions and the coupling conditions play important roles in using the moving plane method. Our results on the symmetry of solutions are further research based on the existence of solutions in [1].

Periodic solutions of fractional Laplace equations: Least period, axial symmetry and limit

We are concerned with periodic solutions of the fractional Laplace equation \begin{equation*} {(-\partial_{xx})^s}u(x)+F'(u(x))=0 \quad \mbox{in }\mathbb{R}, \end{equation*} where $0< s< 1$. The smooth function $F$ is a double-well potential with wells at $+1$ and $-1$. We show that the value of least positive period is $2{\pi}\times({1}/{-F''(0)})^{{1}/({2s})}$. The axial symmetry of odd periodic solutions is obtained by moving plane method. We also prove that odd periodic solutions $u_{T}(x)$ converge to a layer solution of the same equation as periods $T\rightarrow+\infty$.

Symmetry via the moving plane method for a class of quasilinear elliptic problems involving the Hardy potential

We consider positive solutions to a class of quasilinear elliptic problems involving the Hardy potential under zero Dirichlet boundary condition. Via moving plane method, proving a weak comparison principle, we prove symmetry and monotonicity properties for the solutions defined on strictly convex symmetric domains.

Fractional Hardy equations with critical and supercritical exponents

We study the existence, nonexistence and qualitative properties of the solutions to the problem (P)(-Δ)su-θu|x|2s=up-uqinRNu>0inRNu∈H˙s(RN)∩Lq+1(RN),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} ({\\mathcal {P}}) \\quad \\quad \\left\\{ \\begin{aligned} (-\\Delta )^s u -\ heta \\frac{u}{|x|^{2s}}&=u^p - u^q \\quad \ ext {in }\\,\\, {\\mathbb {R}}^N\\\\ u&> 0 \\quad \ ext {in }\\,\\, {\\mathbb {R}}^N\\\\ u&\\in {\\dot{H}}^s({\\mathbb {R}}^N)\\cap L^{q+1}({\\mathbb {R}}^N), \\end{aligned} \\right. \\end{aligned}$$\\end{document}where sin (0,1), N>2s, q>pge {(N+2s)}/{(N-2s)}, theta in (0, Lambda _{N,s}) and Lambda _{N,s} is the sharp constant in the fractional Hardy inequality. For qualitative properties of the solutions, we mean both the radial symmetry, that is obtained by using the moving plane method in a nonlocal setting on the whole mathbb {R}^N, and a suitable upper bound behavior of the solutions. To this last end, we use a representation result that allows us to transform the original problem into a new nonlocal problem in a weighted fractional space.

Regularity and symmetry results for nonlinear degenerate elliptic equations

In this paper we prove regularity results for a class of nonlinear degenerate elliptic equations of the form −div(A(|∇u|)∇u)+B(|∇u|)=f(u); in particular, we investigate the second order regularity of the solutions. As a consequence of these results, we obtain symmetry and monotonicity properties of positive solutions for this class of degenerate problems in convex symmetric domains via a suitable adaption of the celebrated moving plane method of Alexandrov-Serrin.

Monotonicity of positive solutions to quasilinear elliptic equations in half-spaces with a changing-sign nonlinearity

In this paper we prove the monotonicity of positive solutions to \( -\Delta _p u = f(u) \) in half-spaces under zero Dirichlet boundary conditions, for \((2N+2)/(N+2)< p < 2\) and for a general class of regular changing-sign nonlinearities f. The techniques used in the proof of the main result are based on a fine use of comparison and maximum principles and on an adaptation of the celebrated moving plane method to quasilinear elliptic equations in unbounded domains.

Uniqueness of the [formula omitted]-catenary cylinders by their asymptotic behaviour

We establish a uniqueness result for the [φ,e→3]-catenary cylinders by their asymptotic behaviour. Well known examples of such cylinders are the grim reaper translating solitons for the mean curvature flow. For such solitons, F. Martín, J. Pérez-García, A. Savas-Halilaj and K. Smoczyk proved that, if Σ is a properly embedded translating soliton with locally bounded genus and C1-asymptotic to two vertical planes, outside a cylinder, then Σ must coincide with some grim reaper translating soliton. In this paper, applying the moving plane method of Alexandrov together with a strong maximum principle for elliptic operators, we increase the family of [φ,e→3]-minimal graphs where these types of results hold under different assumption of asymptotic behaviour.

Higher order Brezis-Nirenberg problem on hyperbolic spaces: Existence, nonexistence and symmetry of solutions

The main purpose of this paper is to establish the existence, nonexistence and symmetry of nontrivial solutions to the higher order Brezis-Nirenberg problems associated with the GJMS operators Pk on bounded domains in the hyperbolic space Hn and as well as on the entire hyperbolic space Hn. Among techniques different from the higher order Brezis-Nirenberg problem in Euclidean spaces, one of our main novelties in the study of existence and symmetry is to use crucially the Helgason-Fourier analysis on hyperbolic spaces and the higher order Hardy-Sobolev-Maz'ya inequalities and careful study of delicate properties of Green's functions of Pk−λ on hyperbolic spaces which are of independent interests in dealing with such problems. Such Green's functions allow us to obtain the integral representations of solutions and thus avoid using the maximum principle to establish the symmetry of solutions to the higher order Brezis-Nirenberg problem by developing a moving plane method in integral form on hyperbolic spaces in the spirit of the work of [14].

Radial symmetry of solutions to anisotropic and weighted diffusion equations with discontinuous nonlinearities

We prove radial symmetry for bounded nonnegative solutions of a weighted anisotropic problem. Given the anisotropic setting that we deal with, the term "radial" is understood in the Finsler framework. In the whole space, J. Serra obtained the symmetry result in the isotropic unweighted setting. In this case we provide the extension of his result to the anisotropic setting. This provides a generalization to the anisotropic setting of a celebrated result due to Gidas-Ni-Nirenberg and such a generalization is new even for in the case of linear operators whenever the dimension is greater than 2. In proper cones, the results presented are new even in the isotropic and unweighted setting for suitable nonlinear cases. Even for the previously known case of unweighted isotropic setting, the present paper provides an approach to the problem by exploiting integral (in)equalities which is new for $N>2$: this complements the corresponding symmetry result obtained via the moving planes method by Berestycki-Pacella.

Ancient asymptotically cylindrical flows and applications

In this paper, we prove the mean-convex neighborhood conjecture for neck singularities of the mean curvature flow in $${\mathbb {R}}^{n+1}$$ for all $$n\ge 3$$ : we show that if a mean curvature flow $$\{M_t\}$$ in $${\mathbb {R}}^{n+1}$$ has an $$S^{n-1}\times {\mathbb {R}}$$ singularity at $$(x_0,t_0)$$ , then there exists an $$\varepsilon =\varepsilon (x_0,t_0)>0$$ such that $$M_t\cap B(x_0,\varepsilon )$$ is mean-convex for all $$t\in (t_0-\varepsilon ^2,t_0+\varepsilon ^2)$$ . As in the case $$n=2$$ , which was resolved by the first three authors in Choi et al. (Acta Math, 2018), the existence of such a mean-convex neighborhood follows from classifying a certain class of ancient Brakke flows that arise as potential blowup limits near a neck singularity. Specifically, we prove that any ancient unit-regular integral Brakke flow with a cylindrical blowdown must be either a round shrinking cylinder, a translating bowl soliton, or an ancient oval. In particular, combined with a prior result of the last two authors (Hershkovits and White in Commun Pure Appl Math 73(3):558–580, 2020), we obtain uniqueness of mean curvature flow through neck singularities. The main difficulty in addressing the higher dimensional case is in promoting the spectral analysis on the cylinder to global geometric properties of the solution. Most crucially, due to the potential wide variety of self-shrinking flows with entropy lower than the cylinder when $$n\ge 3$$ , smoothness does not follow from the spectral analysis by soft arguments. This precludes the use of the classical moving plane method to derive symmetry. To overcome this, we introduce a novel variant of the moving plane method, which we call “moving plane method without assuming smoothness”—where smoothness and symmetry are established in tandem.

The Moving Plane Method for Doubly Singular Elliptic Equations Involving a First-Order Term

Abstract In this paper we deal with positive singular solutions to semilinear elliptic problems involving a first-order term and a singular nonlinearity. Exploiting a fine adaptation of the well-known moving plane method of Alexandrov–Serrin and a careful choice of the cutoff functions, we deduce symmetry and monotonicity properties of the solutions.

Radial solutions for fully nonlinear elliptic equations of Monge\u2013Amp\xe8re type

First, the symmetry of classical solutions to the Monge–Ampère-type equations is obtained by the moving plane method. Then, the existence and nonexistence of radial solutions in a ball are got from the symmetry results. Finally, the existence and nonexistence of classical solutions to Hessian equations in bounded domains are considered.

On the Symmetry and Recovery of Steady Continuously Stratified Periodic Water Waves

This paper considers two-dimensional steady continuously stratified periodic water waves. We first use the moving plane method to establish that the stratified water waves must be symmetric provided streamlines in a neighborhood of the trough line are monotone. Further, we prove that all streamlines are real analytic (including the free surface) by applying the Schauder estimates on the uniform oblique derivative problem. Finally we provide an analytic expansion method to recover the water waves from relative horizontal velocity on the axis of symmetry and wave height. All results in this paper hold for stratified waves of large (or small) amplitude.