scholarly journals Remarks on radial symmetry and monotonicity for solutions of semilinear higher order elliptic equations

2021 ◽  
Vol 4 (5) ◽  
pp. 1-24
Author(s):  
Filippo Gazzola ◽  
◽  
Gianmarco Sperone ◽  
Keyword(s):  
Unit Ball ◽  
Elliptic Equations ◽  
Biharmonic Equation ◽  
Dirichlet Boundary ◽  
Radial Symmetry ◽  
Dirichlet Boundary Conditions ◽  
Polyharmonic Equations ◽  
Alternative Formula ◽  
Symmetry Properties ◽  
Derivatives Of

<abstract><p>Half a century after the appearance of the celebrated paper by Serrin about overdetermined boundary value problems in potential theory and related symmetry properties, we reconsider semilinear polyharmonic equations under Dirichlet boundary conditions in the unit ball of $ \mathbb{R}^{n} $. We discuss radial properties (symmetry and monotonicity) of positive solutions of such equations and we show that, in <italic>conformal dimensions</italic>, the associated Green function satisfies elegant reflection and symmetry properties related to a suitable Kelvin transform (inversion about a sphere). This yields an alternative formula for computing the partial derivatives of solutions of polyharmonic problems. Moreover, it gives some hints on how to modify a counterexample by Sweers where radial monotonicity fails: we numerically recover strict radial monotonicity for the biharmonic equation in the unit ball of $ \mathbb{R}^{4} $.</p></abstract>

scholarly journals On a singular sublinear polyharmonic problem

2006 ◽  
Vol 2006 ◽  
pp. 1-14 ◽  
Author(s):  
Sonia Ben Othman
Keyword(s):  
Fixed Point ◽  
Green Function ◽  
Fixed Point Theorem ◽  
Boundary Conditions ◽  
Unit Ball ◽  
Existence Result ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Combined Effects ◽  
Polyharmonic Equations

This paper deals with a class of singular nonlinear polyharmonic equations on the unit ballBinℝn (n≥2)where the combined effects of a singular and a sublinear term allow us by using the Schauder fixed point theorem to establish an existence result for the following problem:(−Δ)mu=φ(⋅,u)+ψ(⋅,u)inB(in the sense of distributions),u>0,lim⁡|x|→1u(x)/(1−|x|)m−1=0. Our approach is based on estimates for the polyharmonic Green function onBwith zero Dirichlet boundary conditions.

On the Dispersive Estimate for the Dirichlet Schrödinger Propagator and Applications to Energy Critical NLS

2014 ◽  
Vol 66 (5) ◽  
pp. 1110-1142
Author(s):  
Dong Li ◽  
Guixiang Xu ◽  
Xiaoyi Zhang
Keyword(s):  
Boundary Condition ◽  
Fundamental Solution ◽  
Unit Ball ◽  
Dirichlet Boundary Condition ◽  
Obstacle Problem ◽  
Dirichlet Boundary ◽  
Radial Symmetry ◽  
Well Posedness ◽  
Robust Algorithm ◽  
Dispersive Estimate

AbstractWe consider the obstacle problem for the Schrödinger evolution in the exterior of the unit ball with Dirichlet boundary condition. Under radial symmetry we compute explicitly the fundamental solution for the linear Dirichlet Schrödinger propagator and give a robust algorithm to prove sharp L1 → L∞ dispersive estimates. We showcase the analysis in dimensions n = 5, 7. As an application, we obtain global well–posedness and scattering for defocusing energy-critical NLS on with Dirichlet boundary condition and radial data in these dimensions.

scholarly journals A biharmonic equation with singular nonlinearity

2011 ◽  
Vol 55 (1) ◽  
pp. 155-166 ◽  
Author(s):  
Marius Ghergu
Keyword(s):  
Green Function ◽  
Boundary Conditions ◽  
Bounded Domain ◽  
Existence And Uniqueness ◽  
Biharmonic Equation ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Biharmonic Operator ◽  
The Green Function ◽  
Uniqueness Of A Solution

AbstractWe study the biharmonic equation Δ2u=u−α, 0 < α < 1, in a smooth and bounded domain Ω ⊂ ℝn,n≥ 2, subject to Dirichlet boundary conditions. Under some suitable assumptions on Ω related to the positivity of the Green function for the biharmonic operator, we prove the existence and uniqueness of a solution.

Multiple Solutions for Perturbed Elliptic Equations in Unbounded Domains

2003 ◽  
Vol 3 (1) ◽  
pp. 1-23 ◽  
Author(s):  
A. Salvatore
Keyword(s):  
Schrödinger Equation ◽  
Boundary Conditions ◽  
Perturbation Method ◽  
Elliptic Equations ◽  
Multiple Solutions ◽  
Schrodinger Equation ◽  
Dirichlet Boundary ◽  
Unbounded Domains ◽  
Dirichlet Boundary Conditions

AbstractWe look for solutions of a nonlinear perturbed Schrödinger equation with nonhomogeneous Dirichlet boundary conditions. By using a perturbation method introduced by Bolle, we prove the existence of multiple solutions in spite of the lack of the symmetry of the problem.

scholarly journals Simply Supported Biharmonic Equations with Polynomial Data in The Unit Ball of $ R^n$

2018 ◽  
Author(s):  
Agah D. Garnadi
Keyword(s):  
Unit Ball ◽  
Dirichlet Boundary ◽  
Exact Algorithms ◽  
Dirichlet Boundary Conditions ◽  
Polynomial Functions ◽  
Dirichlet Problems ◽  
Biharmonic Equations ◽  
Simply Supported ◽  
Poisson Equations ◽  
Laplace Equations

We studied simply supported boundary value problem of Biharmonic equation in the unit ball of $R^n, n \geq 3,$ with polynomial data. The problem is restated as a pair of Laplace and Poisson equations with polynomial Dirichlet problems. We utilize an exact algorithms for solving Laplace equations with Dirichlet boundary conditions with polynomial functions data. The algorithm requires only differentiation of the boundary data, but no integration

scholarly journals On a property of the nodal set of least energy sign-changing solutions for quasilinear elliptic equations

2019 ◽  
Vol 149 (5) ◽  
pp. 1163-1173
Author(s):  
Vladimir Bobkov ◽  
Sergey Kolonitskii
Keyword(s):  
Boundary Conditions ◽  
Elliptic Equations ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Quasilinear Elliptic Equations ◽  
Nodal Set ◽  
Sign Changing Solutions ◽  
Quasilinear Elliptic ◽  
Least Energy

AbstractIn this note, we prove the Payne-type conjecture about the behaviour of the nodal set of least energy sign-changing solutions for the equation $-\Delta _p u = f(u)$ in bounded Steiner symmetric domains $ \Omega \subset {{\open R}^N} $ under the zero Dirichlet boundary conditions. The nonlinearity f is assumed to be either superlinear or resonant. In the latter case, least energy sign-changing solutions are second eigenfunctions of the zero Dirichlet p-Laplacian in Ω. We show that the nodal set of any least energy sign-changing solution intersects the boundary of Ω. The proof is based on a moving polarization argument.

Non-existence of solutions for some nonlinear elliptic equations involving measures

2000 ◽  
Vol 130 (1) ◽  
pp. 167-187 ◽  
Author(s):  
L. Orsina ◽  
A. Prignet
Keyword(s):  
Elliptic Equations ◽  
Dirichlet Boundary ◽  
Existence Of Solutions ◽  
Model Problem ◽  
Nonlinear Elliptic Equations ◽  
Dirichlet Boundary Conditions ◽  
Regular Functions ◽  
Nonlinear Elliptic ◽  
Bounded Open Subset ◽  
Image Position

In this paper, we study the non-existence of solutions for the following (model) problem in a bounded open subset Ω of RN: with Dirichlet boundary conditions, where p > 1, q > 1 and μ is a bounded Radon measure. We prove that if λ is a measure which is concentrated on a set of zero r capacity (p < r ≤ N), and if q > r (p − 1)/(r − p), then there is no solution to the above problem, in the sense that if one approximates the measure λ with a sequence of regular functions fn, and if un is the sequence of solutions of the corresponding problems, then un converges to zero.We also study the non-existence of solutions for the bilateral obstacle problem with datum a measure λ concentrated on a set of zero p capacity, with u in for every υ in K, finding again that the only solution obtained by approximation is u = 0.

Padé approximant bounds on the positive solutions of some nonlinear elliptic equations

1978 ◽  
Vol 80 (3-4) ◽  
pp. 183-200
Author(s):  
M. F. Barnsley ◽  
D. Bessis
Keyword(s):  
Laplace Transform ◽  
Elliptic Equations ◽  
Positive Measure ◽  
Dirichlet Boundary ◽  
Homogeneous Equation ◽  
Nonlinear Elliptic Equations ◽  
Dirichlet Boundary Conditions ◽  
Upper And Lower Bounds ◽  
Nonlinear Elliptic ◽  
Positive Functions

SynopsisWe consider the equation Lφ − λpφ + γqφ2 = f on a bounded domain in Rn with homogeneous Neumann-Dirichlet boundary conditions. L is a negative definite uniformly elliptic differential operator, while, p, q and f are positive functions. We show that there exists exactly one positive solution for each λ ∈ R and γ > 0. This solution can be analytically continued throughout Re γ > 0: it is a Laplace transform of a positive measure. The measure is bounded prior to the bifurcation point of the associated “homogeneous” equation and unbounded after. Noting that any Laplace transform of positive measure has associated with it a natural sequence of Tchebycheff systems, it now follows that one can obtain monotonically converging upper and lower bounds which are provided by the generalized Padé approximants generated from the Tchebycheff systems.

Sign in / Sign up

Export Citation Format

Share Document