DOAJ (DOAJ: Directory of Open Access Journals), May 1, 2017
In this paper, we study a coupled system of semilinear Riemann-Liouville type fractional differen... more In this paper, we study a coupled system of semilinear Riemann-Liouville type fractional differential equations. Applying the Schäuder fixed point theorem, we prove the existence of positive solutions of the fractional differential system. Furthermore, the asymptotic behavior of the solutions is given by using Karamata regular variation theory.
In this article, we take up the existence and the asymptotic behavior of entire bounded positive ... more In this article, we take up the existence and the asymptotic behavior of entire bounded positive solutions to the following semilinear elliptic system: −∆u = a (x)u α v r , x ∈ ℝ n (n ≥), −∆v = a (x)v
ABSTRACT La perturbation semi-linaire des rsolvantes et des semi-groupes linaires, nous donne des... more ABSTRACT La perturbation semi-linaire des rsolvantes et des semi-groupes linaires, nous donne des rsolvantes et des semi-groupes non linaires. Nous tudions alors les proprits de ces oprateurs non linaires et en particulier les fonctions surmdianes et excessives associes.We are concerned with nonlinear resolvents and semi-groups. They are obtained by perturbing linear ones. Properties of these nonlinear operators are investigated, particularly supermedian and excessive functions.
Calculus of Variations and Partial Differential Equations, 2020
In this paper, we consider the asymptotic behavior of positive solutions of the biharmonic equati... more In this paper, we consider the asymptotic behavior of positive solutions of the biharmonic equation ∆ 2 u = u p in B 1 \{0} with an isolated singularity, where the punctured ball B 1 \{0} ⊂ R n with n ≥ 5 and n n−4 < p < n+4 n−4. This equation is relevant for the Q-curvature problem in conformal geometry. We classify isolated singularities of positive solutions and describe the asymptotic behavior of positive singular solutions without the sign assumption for −∆u. We also give a new method to prove removable singularity theorem for nonlinear higher order equations.
The aim of this paper is to establish existence and uniqueness of a positive continuous solution ... more The aim of this paper is to establish existence and uniqueness of a positive continuous solution to the following singular nonlinear problem. {− 1− (−1) = () , ∈ (0, 1), lim →0 −1 () = 0, (1) = 0}, where ≥ 3, < 1, and denotes a nonnegative continuous function that might have the property of being singular at = 0 and /or = 1 and which satisfies certain condition associated to Karamata class. We emphasize that the nonlinearity might also be singular at = 0, while the solution could blow-up at 0. Our method is based on the global estimates of potential functions and the Schauder fixed point theorem.
We study the following fractional Navier boundary value problem: ⎧ ⎪ ⎨ ⎪ ⎩ D α (D β u)(x) + u(x)f... more We study the following fractional Navier boundary value problem: ⎧ ⎪ ⎨ ⎪ ⎩ D α (D β u)(x) + u(x)f (x, u(x)) = 0, 0 < x < 1, lim x→0 + D β-1 u(x) = 0, lim x→0 + D α-1 (D β u)(x) = ξ , u(1) = 0, D β u(1) =-ζ , where α, β ∈ (1, 2], D α and D β stand for the standard Riemann-Liouville fractional derivatives, and ξ , ζ ≥ 0 are such that ξ + ζ > 0. Our purpose is to prove the existence, uniqueness, and global asymptotic behavior of a positive continuous solution, where f : (0, 1) × [0, ∞) → [0, ∞) is continuous and dominated by a function p satisfying appropriate integrability condition.
Communications in Contemporary Mathematics, Jun 1, 2003
We establish a 3G-Theorem for the Green's function for an unbounded regular domain D in ℝn(n ... more We establish a 3G-Theorem for the Green's 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 [Formula: see text]. Next, we study the existence and the uniqueness of a positive continuous solution u in [Formula: see text] of the following nonlinear singular elliptic problem [Formula: see text] 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.
We are concerned with the existence, uniqueness and global asymptotic behavior of positive contin... more We are concerned with the existence, uniqueness and global asymptotic behavior of positive continuous solutions to the second-order boundary value problem A (Au ὔ) ὔ + a (t)u σ + a (t)u σ = , t ∈ (, ∞), subject to the boundary conditions lim t→ + u(t) = , lim t→∞ u(t)/ρ(t) = , where σ , σ < and A is a continuous function on [ , ∞) which is positive and di erentiable on (, ∞) such that ∫ /A(t) dt < ∞ and ∫ ∞ /A(t) dt = ∞. Here, ρ(t) = ∫ t /A(s) ds for t > and a , a are nonnegative continuous functions on (, ∞) that may be singular at t = and satisfying some appropriate assumptions related to the Karamata regular variation theory. Our approach is based on the sub-supersolution method.
In this paper, we study the existence and nonexistence of positive bounded solutions of the integ... more In this paper, we study the existence and nonexistence of positive bounded solutions of the integral equation u = λV(af (u)), where λ is a positive parameter, a is a nontrivial nonnegative measurable function with bounded potential and V belongs to a class of positive kernels that contains in particular the potential kernel of the classical Laplacian V = (-)-1 or V = (∂ ∂t-)-1 or the inverse of the polyharmonic Laplacian (-) m , m ≥ 2.
Journal of Mathematical Analysis and Applications, Sep 1, 2007
We study the nonlinear parabolic equation u − uϕ(., u) − ∂u ∂t = 0, in R n × (0, ∞) with boundary... more We study the nonlinear parabolic equation u − uϕ(., u) − ∂u ∂t = 0, in R n × (0, ∞) with boundary condition u(x, 0) = u 0 (x), not necessarily bounded function. The nonlinearity ϕ((x, t), u) is required to satisfy some conditions related to the parabolic Kato class P ∞ (R n) while allowing existence of positive solutions of the equation and continuity of such solutions. Our approach is based on potential theory tools.
Journal of Mathematical Analysis and Applications, Sep 1, 2010
Let Ω be a C 1,1-bounded domain in R n for n 2. In this paper, we are concerned with the asymptot... more Let Ω be a C 1,1-bounded domain in R n for n 2. In this paper, we are concerned with the asymptotic behavior of the unique positive classical solution to the singular boundary-value problem u + a(x)u −σ = 0 in Ω, u |∂Ω = 0, where σ 0, a is a nonnegative function in C α loc (Ω), 0 < α < 1 and there exists c > 0 such that 1 c a(x)(δ(x)) λ m k=1 (Log k (ω δ(x))) μ k c. Here λ 2, μ k ∈ R, ω is a positive constant and δ(x) = dist(x, ∂Ω).
This paper deals with the following boundary value problem D α uðtÞ = f ðt, uðtÞÞ, t ∈ ð0, 1Þ, uð... more This paper deals with the following boundary value problem D α uðtÞ = f ðt, uðtÞÞ, t ∈ ð0, 1Þ, uð0Þ = uð1Þ = D α−3 uð0Þ = u′ð1Þ = 0, (where 3 < α ≤ 4, D α is the Riemann-Liouville fractional derivative, and the nonlinearity f , which could be singular at both t = 0 and t = 1, is required to be continuous on ð0, 1Þ × ℝ satisfying a mild Lipschitz assumption. Based on the Banach fixed point theorem on an appropriate space, we prove that this problem possesses a unique continuous solution u satisfying juðtÞj ≤ cωðtÞ, for t ∈ ½0, 1 and c > 0, where ωðtÞ ≔ t α−2 ð1 − tÞ 2 :
We deal with the following Riemann-Liouville fractional nonlinear boundary value problem: D α v(x... more We deal with the following Riemann-Liouville fractional nonlinear boundary value problem: D α v(x) + f (x, v(x)) = 0, 2 < α ≤ 3, x ∈ (0, 1), v(0) = v (0) = v(1) = 0. Under mild assumptions, we prove the existence of a unique continuous solution v to this problem satisfying v(x) ≤ cx α-1 (1-x) for all x ∈ [0, 1] and some c > 0.
Fractional Calculus and Applied Analysis, Aug 1, 2016
Using estimates on the Green function and a perturbation argument, we prove the existence and uni... more Using estimates on the Green function and a perturbation argument, we prove the existence and uniqueness of a positive continuous solution to problem:
DOAJ (DOAJ: Directory of Open Access Journals), Apr 1, 2016
In this article, we study the superlinear fractional boundary-value problem D α u(x) = u(x)g(x, u... more In this article, we study the superlinear fractional boundary-value problem D α u(x) = u(x)g(x, u(x)), 0 < x < 1, u(0) = 0, lim x→0 + D α−3 u(x) = 0, lim x→0 + D α−2 u(x) = ξ, u (1) = ζ, where 3 < α ≤ 4, D α is the Riemann-Liouville fractional derivative and ξ, ζ ≥ 0 are such that ξ + ζ > 0. The function g(x, u) ∈ C((0, 1) × [0, ∞), [0, ∞)) that may be singular at x = 0 and x = 1 is required to satisfy convenient hypotheses to be stated later. By means of a perturbation argument, we establish the existence, uniqueness and global asymptotic behavior of a positive continuous solution to the above problem.An example is given to illustrate our main results.
DOAJ (DOAJ: Directory of Open Access Journals), Jun 1, 2015
In this article, we study the existence and nonexistence of positive bounded solutions of the Dir... more In this article, we study the existence and nonexistence of positive bounded solutions of the Dirichlet problem −∆u = λp(x)f (u, v), in R n + , −∆v = λq(x)g(u, v), in R n + , u = v = 0 on ∂R n + , lim |x|→∞ u(x) = lim |x|→∞ v(x) = 0, where R n + = {x = (x 1 , x 2 ,. .. , xn) ∈ R n : xn > 0} (n ≥ 3) is the upper half-space and λ is a positive parameter. The potential functions p, q are not necessarily bounded, they may change sign and the functions f, g : R 2 → R are continuous. By applying the Leray-Schauder fixed point theorem, we establish the existence of positive solutions for λ sufficiently small when f (0, 0) > 0 and g(0, 0) > 0. Some nonexistence results of positive bounded solutions are also given either if λ is sufficiently small or if λ is large enough.
This work deals with the existence of positive solutions of convection-diffusion equations ∆u + f... more This work deals with the existence of positive solutions of convection-diffusion equations ∆u + f (x, u, ∇u) = 0 in an exterior domain of R n (n ≥ 3).
The aim of this paper is to establish the existence and global asymptotic behavior of a positive ... more The aim of this paper is to establish the existence and global asymptotic behavior of a positive continuous solution for the following semilinear problem: ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ u(x) = a(x)u σ (x), x ∈ D, u > 0, in D, u(x) = 0, x ∈ ∂D, lim |x|→∞ u(x) ln |x| = 0, where σ < 1, D is an unbounded domain in R 2 with a compact nonempty boundary ∂D consisting of finitely many Jordan curves. As main tools, we use Kato class, Karamata regular variation theory and the Schauder fixed point theorem.
DOAJ (DOAJ: Directory of Open Access Journals), May 1, 2017
In this paper, we study a coupled system of semilinear Riemann-Liouville type fractional differen... more In this paper, we study a coupled system of semilinear Riemann-Liouville type fractional differential equations. Applying the Schäuder fixed point theorem, we prove the existence of positive solutions of the fractional differential system. Furthermore, the asymptotic behavior of the solutions is given by using Karamata regular variation theory.
In this article, we take up the existence and the asymptotic behavior of entire bounded positive ... more In this article, we take up the existence and the asymptotic behavior of entire bounded positive solutions to the following semilinear elliptic system: −∆u = a (x)u α v r , x ∈ ℝ n (n ≥), −∆v = a (x)v
ABSTRACT La perturbation semi-linaire des rsolvantes et des semi-groupes linaires, nous donne des... more ABSTRACT La perturbation semi-linaire des rsolvantes et des semi-groupes linaires, nous donne des rsolvantes et des semi-groupes non linaires. Nous tudions alors les proprits de ces oprateurs non linaires et en particulier les fonctions surmdianes et excessives associes.We are concerned with nonlinear resolvents and semi-groups. They are obtained by perturbing linear ones. Properties of these nonlinear operators are investigated, particularly supermedian and excessive functions.
Calculus of Variations and Partial Differential Equations, 2020
In this paper, we consider the asymptotic behavior of positive solutions of the biharmonic equati... more In this paper, we consider the asymptotic behavior of positive solutions of the biharmonic equation ∆ 2 u = u p in B 1 \{0} with an isolated singularity, where the punctured ball B 1 \{0} ⊂ R n with n ≥ 5 and n n−4 < p < n+4 n−4. This equation is relevant for the Q-curvature problem in conformal geometry. We classify isolated singularities of positive solutions and describe the asymptotic behavior of positive singular solutions without the sign assumption for −∆u. We also give a new method to prove removable singularity theorem for nonlinear higher order equations.
The aim of this paper is to establish existence and uniqueness of a positive continuous solution ... more The aim of this paper is to establish existence and uniqueness of a positive continuous solution to the following singular nonlinear problem. {− 1− (−1) = () , ∈ (0, 1), lim →0 −1 () = 0, (1) = 0}, where ≥ 3, < 1, and denotes a nonnegative continuous function that might have the property of being singular at = 0 and /or = 1 and which satisfies certain condition associated to Karamata class. We emphasize that the nonlinearity might also be singular at = 0, while the solution could blow-up at 0. Our method is based on the global estimates of potential functions and the Schauder fixed point theorem.
We study the following fractional Navier boundary value problem: ⎧ ⎪ ⎨ ⎪ ⎩ D α (D β u)(x) + u(x)f... more We study the following fractional Navier boundary value problem: ⎧ ⎪ ⎨ ⎪ ⎩ D α (D β u)(x) + u(x)f (x, u(x)) = 0, 0 < x < 1, lim x→0 + D β-1 u(x) = 0, lim x→0 + D α-1 (D β u)(x) = ξ , u(1) = 0, D β u(1) =-ζ , where α, β ∈ (1, 2], D α and D β stand for the standard Riemann-Liouville fractional derivatives, and ξ , ζ ≥ 0 are such that ξ + ζ > 0. Our purpose is to prove the existence, uniqueness, and global asymptotic behavior of a positive continuous solution, where f : (0, 1) × [0, ∞) → [0, ∞) is continuous and dominated by a function p satisfying appropriate integrability condition.
Communications in Contemporary Mathematics, Jun 1, 2003
We establish a 3G-Theorem for the Green's function for an unbounded regular domain D in ℝn(n ... more We establish a 3G-Theorem for the Green's 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 [Formula: see text]. Next, we study the existence and the uniqueness of a positive continuous solution u in [Formula: see text] of the following nonlinear singular elliptic problem [Formula: see text] 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.
We are concerned with the existence, uniqueness and global asymptotic behavior of positive contin... more We are concerned with the existence, uniqueness and global asymptotic behavior of positive continuous solutions to the second-order boundary value problem A (Au ὔ) ὔ + a (t)u σ + a (t)u σ = , t ∈ (, ∞), subject to the boundary conditions lim t→ + u(t) = , lim t→∞ u(t)/ρ(t) = , where σ , σ < and A is a continuous function on [ , ∞) which is positive and di erentiable on (, ∞) such that ∫ /A(t) dt < ∞ and ∫ ∞ /A(t) dt = ∞. Here, ρ(t) = ∫ t /A(s) ds for t > and a , a are nonnegative continuous functions on (, ∞) that may be singular at t = and satisfying some appropriate assumptions related to the Karamata regular variation theory. Our approach is based on the sub-supersolution method.
In this paper, we study the existence and nonexistence of positive bounded solutions of the integ... more In this paper, we study the existence and nonexistence of positive bounded solutions of the integral equation u = λV(af (u)), where λ is a positive parameter, a is a nontrivial nonnegative measurable function with bounded potential and V belongs to a class of positive kernels that contains in particular the potential kernel of the classical Laplacian V = (-)-1 or V = (∂ ∂t-)-1 or the inverse of the polyharmonic Laplacian (-) m , m ≥ 2.
Journal of Mathematical Analysis and Applications, Sep 1, 2007
We study the nonlinear parabolic equation u − uϕ(., u) − ∂u ∂t = 0, in R n × (0, ∞) with boundary... more We study the nonlinear parabolic equation u − uϕ(., u) − ∂u ∂t = 0, in R n × (0, ∞) with boundary condition u(x, 0) = u 0 (x), not necessarily bounded function. The nonlinearity ϕ((x, t), u) is required to satisfy some conditions related to the parabolic Kato class P ∞ (R n) while allowing existence of positive solutions of the equation and continuity of such solutions. Our approach is based on potential theory tools.
Journal of Mathematical Analysis and Applications, Sep 1, 2010
Let Ω be a C 1,1-bounded domain in R n for n 2. In this paper, we are concerned with the asymptot... more Let Ω be a C 1,1-bounded domain in R n for n 2. In this paper, we are concerned with the asymptotic behavior of the unique positive classical solution to the singular boundary-value problem u + a(x)u −σ = 0 in Ω, u |∂Ω = 0, where σ 0, a is a nonnegative function in C α loc (Ω), 0 < α < 1 and there exists c > 0 such that 1 c a(x)(δ(x)) λ m k=1 (Log k (ω δ(x))) μ k c. Here λ 2, μ k ∈ R, ω is a positive constant and δ(x) = dist(x, ∂Ω).
This paper deals with the following boundary value problem D α uðtÞ = f ðt, uðtÞÞ, t ∈ ð0, 1Þ, uð... more This paper deals with the following boundary value problem D α uðtÞ = f ðt, uðtÞÞ, t ∈ ð0, 1Þ, uð0Þ = uð1Þ = D α−3 uð0Þ = u′ð1Þ = 0, (where 3 < α ≤ 4, D α is the Riemann-Liouville fractional derivative, and the nonlinearity f , which could be singular at both t = 0 and t = 1, is required to be continuous on ð0, 1Þ × ℝ satisfying a mild Lipschitz assumption. Based on the Banach fixed point theorem on an appropriate space, we prove that this problem possesses a unique continuous solution u satisfying juðtÞj ≤ cωðtÞ, for t ∈ ½0, 1 and c > 0, where ωðtÞ ≔ t α−2 ð1 − tÞ 2 :
We deal with the following Riemann-Liouville fractional nonlinear boundary value problem: D α v(x... more We deal with the following Riemann-Liouville fractional nonlinear boundary value problem: D α v(x) + f (x, v(x)) = 0, 2 < α ≤ 3, x ∈ (0, 1), v(0) = v (0) = v(1) = 0. Under mild assumptions, we prove the existence of a unique continuous solution v to this problem satisfying v(x) ≤ cx α-1 (1-x) for all x ∈ [0, 1] and some c > 0.
Fractional Calculus and Applied Analysis, Aug 1, 2016
Using estimates on the Green function and a perturbation argument, we prove the existence and uni... more Using estimates on the Green function and a perturbation argument, we prove the existence and uniqueness of a positive continuous solution to problem:
DOAJ (DOAJ: Directory of Open Access Journals), Apr 1, 2016
In this article, we study the superlinear fractional boundary-value problem D α u(x) = u(x)g(x, u... more In this article, we study the superlinear fractional boundary-value problem D α u(x) = u(x)g(x, u(x)), 0 < x < 1, u(0) = 0, lim x→0 + D α−3 u(x) = 0, lim x→0 + D α−2 u(x) = ξ, u (1) = ζ, where 3 < α ≤ 4, D α is the Riemann-Liouville fractional derivative and ξ, ζ ≥ 0 are such that ξ + ζ > 0. The function g(x, u) ∈ C((0, 1) × [0, ∞), [0, ∞)) that may be singular at x = 0 and x = 1 is required to satisfy convenient hypotheses to be stated later. By means of a perturbation argument, we establish the existence, uniqueness and global asymptotic behavior of a positive continuous solution to the above problem.An example is given to illustrate our main results.
DOAJ (DOAJ: Directory of Open Access Journals), Jun 1, 2015
In this article, we study the existence and nonexistence of positive bounded solutions of the Dir... more In this article, we study the existence and nonexistence of positive bounded solutions of the Dirichlet problem −∆u = λp(x)f (u, v), in R n + , −∆v = λq(x)g(u, v), in R n + , u = v = 0 on ∂R n + , lim |x|→∞ u(x) = lim |x|→∞ v(x) = 0, where R n + = {x = (x 1 , x 2 ,. .. , xn) ∈ R n : xn > 0} (n ≥ 3) is the upper half-space and λ is a positive parameter. The potential functions p, q are not necessarily bounded, they may change sign and the functions f, g : R 2 → R are continuous. By applying the Leray-Schauder fixed point theorem, we establish the existence of positive solutions for λ sufficiently small when f (0, 0) > 0 and g(0, 0) > 0. Some nonexistence results of positive bounded solutions are also given either if λ is sufficiently small or if λ is large enough.
This work deals with the existence of positive solutions of convection-diffusion equations ∆u + f... more This work deals with the existence of positive solutions of convection-diffusion equations ∆u + f (x, u, ∇u) = 0 in an exterior domain of R n (n ≥ 3).
The aim of this paper is to establish the existence and global asymptotic behavior of a positive ... more The aim of this paper is to establish the existence and global asymptotic behavior of a positive continuous solution for the following semilinear problem: ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ u(x) = a(x)u σ (x), x ∈ D, u > 0, in D, u(x) = 0, x ∈ ∂D, lim |x|→∞ u(x) ln |x| = 0, where σ < 1, D is an unbounded domain in R 2 with a compact nonempty boundary ∂D consisting of finitely many Jordan curves. As main tools, we use Kato class, Karamata regular variation theory and the Schauder fixed point theorem.
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Papers by Habib Maagli