Class Of Singular Elliptic Equations Research Articles

Modulus of continuity of weak solutions to a class of singular elliptic equations

Modulus of continuity of weak solutions to a class of singular elliptic equations

New techniques for solving some class of singular elliptic equations

New techniques for solving some class of singular elliptic equations

Multiplicity of positive solutions for a class of singular elliptic equations with critical Sobolev exponent and Kirchhoff-type nonlocal term

Multiplicity of positive solutions for a class of singular elliptic equations with critical Sobolev exponent and Kirchhoff-type nonlocal term

Existence of Weak Solutions to a Class of Singular Elliptic Equations

This paper is concerned with the existence of solutions to the following singular elliptic boundary value problem involving \({p}\)-Laplace operator $$-{\rm div}(|\nabla u|^{p-2} \nabla u)=\frac{h}{u^\gamma} \text{in } \Omega, \quad u > 0\text{ in } \Omega,\quad u=0 \text{ on } \partial\Omega.$$ Here, \({\Omega\subset \mathbb{R}^N(N\geq3)}\) is a bounded domain with smooth boundary, and \({h}\) is a positive \({L^1}\) function on \({\Omega}\). A “compatibility condition” on the couple \({(h(x),\gamma)}\) is given for the problem to have at least one solution. More precisely, it is shown that the problem admits at least one solution if and only if there exists a \({u_0\in W_0^{1,p}(\Omega)}\) such that \({\int_\Omega hu_0^{1-\gamma} \mathrm{d}x < \infty}\). This generalizes a previous result obtained by Sun and Zhang (Calc Var Partial Differ Equ 49:909–922, 2014) who considered the case \({p=2}\).

Existence of positive weak solutions for a class of singular elliptic equations

In this note, we are concerned with positive solutions for a class of singular elliptic equations. Under some conditions, we obtain weak solutions for the equations by elliptic regularization method and sub-super solution method.

Weighted Sobolev's inequalities for bounded domains and singular elliptic equations

Using a method developed by P´ erez and Wheeden and the representation of smooth functions by integral operators whose kernels are gradients of the Green functions, we obtain weighted Sobolev's inequalities for bounded domains which im- prove and unify several kinds of inequalities. From these results we establish Green functions and the existence, uniqueness and regularity results for a class of singular elliptic equations.

Semi-linear singular elliptic equations with dependence on the gradient

We establish the existence of a positive solution for the following non-variational equation { − div ( | x | − 2 a ∇ u ) = | x | − 2 ( a + 1 ) + c f ( x , u , ∇ u ) , in Ω u = 0 , on ∂ Ω , where the non-linearity f ( x , t , ξ ) belongs to a class of functions that are superlinear in the variable t and sublinear in the variable ξ . For this purpose we used an idea of a recent work by De Figueiredo et al. [D. De Figueiredo, M. Girardi, M. Matzeu, Semilinear elliptic equations with dependence on the gradient via mountain-pass techniques, Diff. Integral Equ. (in press)] and we established a new regularity result for a class of Singular Elliptic Equations.

Resonance Problems for a Class of Singular Elliptic Equations

In this article we study the resonance problems for Hardy-Sobolev operator $ L_{\\mu }:= -\\Delta _{p}-\\fraca {(\\mu }{|x|^{p})}, 0\\le \\mu \\lt {\\fraca {(N-p}{p} )}\\,^{p} $ with unbounded nonlinearity. Existence of weak solutions is established using the minimax theorems in critical point theory.

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.

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&gt;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.