Imed Bachar

Estimates on the Green Function and Existence of Positive Solutions of Nonlinear Singular Elliptic Equations

We establish a 3G-Theorem for the Greens function for an unbounded regular domain D in ℝn(n ≥ 3), with compact boundary. We exploit this result to introduce a new class of potentials K(D) that properly contains the classical Kato class . Next, we study the existence and the uniqueness of a positive continuous solution u in of the following nonlinear singular elliptic problem where φ is a nonnegative Borel measurable function in D × (0, ∞), that belongs to a convex cone which contains, in particular, all functions φ(x, t) = q(x)t-σ, σ ≥ 0 with q ∈ K(D). We give also some estimates on the solution u.

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

We prove some existence and nonexistence results for the semilinear elliptic equation Δu = q(x)f(u) on Ω ⊆ Rn (n ≥ 2) where u is required to blow up on the boundary of Ω and f is a nonnegative function which is assumed to be Lipschitz continuous and bounded away from zero on each interval [e, ∞) and have at worst linear growth.In particular, we extend some results already obtained in the case where f(u)=uγ, 0 > γ ≤ 1.

Estimates for the Green function and singular solutions for polyharmonic nonlinear equation

We establish a new form of the 3G theorem for polyharmonic Green function on the unit ball of ℝn(n≥2) corresponding to zero Dirichlet boundary conditions. This enables us to introduce a new class of functions Km,n containing properly the classical Kato class Kn. We exploit properties of functions belonging to Km,n to prove an infinite existence result of singular positive solutions for nonlinear elliptic equation of order 2m.

Existence and global asymptotic behavior of positive solutions for combined second-order differential equations on the half-line

Abstract We are concerned with the existence, uniqueness and global asymptotic behavior of positive continuous solutions to the second-order boundary value problem 1 A ⁢ ( A ⁢ u ′ ) ′ + a 1 ⁢ ( t ) ⁢ u σ 1 + a 2 ⁢ ( t ) ⁢ u σ 2 = 0 , t ∈ ( 0 , ∞ ) ,

ESTIMATES ON THE GREEN FUNCTION AND EXISTENCE OF POSITIVE SOLUTIONS FOR SOME NONLINEAR POLYHARMONIC PROBLEMS OUTSIDE THE UNIT BALL

\frac{1}{A}(Au^{\prime})^{\prime}+a_{1}(t)u^{\sigma_{1}}+a_{2}(t)u^{\sigma_{2}% }=0,\quad t\in(0,\infty),

On a nonlinear singular one-dimensional parabolic equation and double Laplace decomposition method:

subject to the boundary conditions lim t → 0 + ⁡ u ⁢ ( t ) = 0

Modified Sumudu decomposition method for Lane-Emden-type differential equations

{\lim_{t\rightarrow 0^{+}}u(t)=0}

Existence of Positive Solutions for Some Superlinear Fourth-Order Boundary Value Problems

, lim t → ∞ ⁡ u ⁢ ( t ) / ρ ⁢ ( t ) = 0

Estimates for the green function and existence of positive solutions for higher-order elliptic equations

{\lim_{t\rightarrow\infty}{u(t)}/{\rho(t)}=0}

Sharp estimates on the solutions to combined fractional boundary value problems on the half-line

, where σ 1 , σ 2 < 1