Singular Elliptic Equations Research Articles

On the existence of positive solutions for a class of singular elliptic equations

We consider the class of equations $$ -\Delta u={A\over {|x|^{\alpha}}}u+u^{\theta} \qquad\qquad x\in\Bbb R^n\setminus\{0\} $$ where $A\in \Bbb R$, $\alpha>0$ and $\theta>1$. Depending on the values of the three parameters involved, we obtain both results of existence and nonexistence of positive solutions by combining the moving planes and the moving spheres methods through the Kelvin's inversion map and classical arguments on ODE's.

On a class of singular quasilinear elliptic equations with general structure and distribution data

On a class of singular quasilinear elliptic equations with general structure and distribution data

Classical radial solutions for singular and degenerate elliptic equations

Classical radial solutions for singular and degenerate elliptic equations

A theorem on global regularity of solutions of a singular elliptic equation

Let u be a solution of a singular elliptic equation Δu + bur + f ≡ 0 where b has a first order singularity on the outer boundary ∂+ of an annular domain, and assume that u and ∞ satisfy certain decay conditions near ∂+. Then a regularity property of u near ∂+ is proved, provided the residue of b at ∂+ is large enough

Existence results for singular elliptic equations

Existence results for singular elliptic equations

Generalized solution of a Dirichlet-type problem for some singular elliptic equations

Generalized solution of a Dirichlet-type problem for some singular elliptic equations

Radially symmetric solutions of a class of singular elliptic equations

The boundary value problemis studied with a view to obtaining the existence of positive solutions in C1([0, 1])∩C2((0, 1)). The function f is assumed to be singular in the second variable, with the singularity modeled after the special case f(x, y) = a(x)y−p, p>0.This boundary value problem arises in the search of positive radially symmetric solutions towhere Ω is the open unit ball in ℝN, centered at the origin, Γ is its boundary and |x| is the Euclidean norm of x.

Finite element method (FEM) for a singular elliptic equation

Finite element method (FEM) for a singular elliptic equation

Nonlinear Singular Elliptic Equations

We improve the Poincaré inequality, the Sobolev embedding theorem and the Trudinger embedding theorem and prove a mountain-pass theorem. Applying these results we study a nonlinear singular mixed boundary problem.

Entire solutions of singular elliptic equations

On etudie les solutions entieres positives de l'equation elliptique singuliere −Δu=f(x)u − λ, λ>0, ou f∈C loc α(R n ), 0 0 pour |x|>0

A class of strongly degenerate elliptic operators

Using a weighted Poincaré inequality, we study (ω1,…,ωn)-elliptic operators. This method is applied to solve singular elliptic equations with boundary conditions in W1,2. We also obtain a result about the regularity of solutions of singular elliptic equations. An application to (ω1,…,ωn)-parabolic equations is given.

On a Singular Elliptic Equation

In this paper, we study the singular elliptic equation $Lu + K(x){u^p} = 0$, where $L$ is a uniformly elliptic operator of divergence form, $p > 1$ and $K(x)$ has a singularity at the origin. We prove that this equation does not possess any positive (local) solution in any punctured neighborhood of the origin if there exist two constants ${C_1}$, ${C_2}$ such that ${C_1}|x{|^\sigma } \geqslant K(x) \geqslant {C_2}|x{|^\sigma }$ near the origin for some $\sigma \leqslant - 2$ (with no other condition on the gradient of $K$ ). In fact, an integral condition is derived.

On a singular elliptic equation

In this paper, we study the singular elliptic equation L u + K ( x ) u p = 0 Lu + K(x){u^p} = 0 , where L L is a uniformly elliptic operator of divergence form, p > 1 p > 1 and K ( x ) K(x) has a singularity at the origin. We prove that this equation does not possess any positive (local) solution in any punctured neighborhood of the origin if there exist two constants C 1 {C_1} , C 2 {C_2} such that C 1 | x | σ ⩾ K ( x ) ⩾ C 2 | x | σ {C_1}|x{|^\sigma } \geqslant K(x) \geqslant {C_2}|x{|^\sigma } near the origin for some σ ⩽ − 2 \sigma \leqslant - 2 (with no other condition on the gradient of K K ). In fact, an integral condition is derived.

A Regularity Theorem for a Singular Elliptic Equation

We consider an elliptic equation whose coefficients become singular at an isolated point and show that a weak solution belonging to a certain weighted Sobolev space must have a removable singularity at that point. AMS(MOS): 35B65, 35D10, 35J70

O(3)-symmetric merons inSU(N)gauge theory

We investigate nonreducible O(3)-symmetric meron solutions to classical $\mathrm{SU}(N+1)$ Yang-Mills theory in four-dimensional Euclidean space. For even $N$ the solutions have topological charge densities equal to a sum of $\ensuremath{\delta}$ functions with integer coefficients while for odd $N(Ng1)$ these coefficients can be both integer and half-integer. In all cases they correspond to solutions of a system of $N$ coupled singular elliptic equations. We discuss the existence of two-meron solutions of this system and for $N=3,4$ give some numerical solutions.

O(3) symmetric merons in an SU(3) Yang-Mills theory

We investigate irreducible, O(3) symmetric multiple-meron solutions to the classical SU(3) Yang-Mills equations in four-dimensional Euclidean space. The solutions have topological charge density equal to a sum of delta-functions with integer coefficients, and correspond to solutions of a system of two coupled singular elliptic equations. We prove the existence of twomeron solutions of the coupled system.

The Dirichlet problem for a singular elliptic equation

The Dirichlet problem for a singular elliptic equation

Dirichlet Problems for Singular Elliptic Equations

Dirichlet Problems for Singular Elliptic Equations

Dirichlet problems for singular elliptic equations

Boundary value problems are formulated for the equation \[ ( ∗ ) L [ u ] = ∑ i , j = 1 n a i j ∂ 2 u ∂ x i ∂ x j + ∑ i = 1 n − 1 b i ∂ u ∂ x i + h x n ∂ u ∂ x n + c u = f ( \ast )\quad L[u] = \sum \limits _{i,j = 1}^n {{a_{ij}}\frac {{{\partial ^2}u}}{{\partial {x_i}\partial {x_j}}}} + \sum \limits _{i = 1}^{n - 1} {{b_i}\frac {{\partial u}}{{\partial {x_i}}}} + \frac {h}{{{x_n}}}\frac {{\partial u}}{{\partial {x_n}}} + cu = f \] in a bounded domain G G in E n {E_n} with boundary ∂ G = S 1 ∪ S 2 \partial G = {S_1} \cup {S_2} where S 1 {S_1} is in x n = 0 {x_n} = 0 and S 2 {S_2} is in x n > 0 {x_n} > 0 . A uniqueness theorem is established for ( ∗ ) ( \ast ) when boundary data is only given on S 2 {S_2} for \[ h ( x 1 , ⋯ , x n − 1 , 0 ) ≧ 1 ; h({x_1}, \cdots ,{x_{n - 1}},0) \geqq 1; \] ; whereas an existence and uniqueness theorem for the Dirichlet problem is proved for h ( x 1 , x 2 , ⋯ , x n − 1 , 0 ) > 1 h({x_1},{x_2}, \cdots ,{x_{n - 1}},0) > 1 .

Boundary value problems of singular elliptic partial differential equations

In a recent paper [6], this author has extended the method of the kernel function [1] to the boundary value problems of the generalized axially symmetric potentialsThis method can also be applied to a more general class of singular differential equations, namelyor, equivalently,We shall derive in the sequel explicit formulas for the Dirichlet problems of (1.1) in the first quadrant of the x-y plane in terms of sufficiently smooth boundary data, and obtain an error-bound for their approximate solutions. We shall also indicate how the Neumann problem can be solved.