Haydar Abdel Hamid

Bio: Haydar Abdel Hamid is an academic researcher from Centre national de la recherche scientifique. The author has contributed to research in topics: Measure (mathematics) & Connection (algebraic framework). The author has an hindex of 4, co-authored 4 publications receiving 99 citations.

On the connection between two quasilinear elliptic problems with source terms of order 0 or 1

Abstract: We establish a precise connection between two elliptic quasilinear problems with Dirichlet data in a bounded domain of ℝN. The first one, of the form involves a source gradient term with natural growth, where β is non-negative, λ > 0, f(x) ≧ 0, and α is a non-negative measure. The second one, of the form presents a source term of order 0, where g is non-decreasing, and μ is a non-negative measure. Here β and g can present an asymptote. The correlation gives new results of existence, non-existence, regularity and multiplicity of the solutions for the two problems, without or with measures. New informations on the extremal solutions are given when g is superlinear.

On the connection between two quasilinear elliptic problems with source terms of order 0 or 1

Abstract: We establish a precise connection between two elliptic quasilinear problems with Dirichlet data in a bounded domain of $\mathbb{R}^{N}.$ The first one, of the form \[ -\Delta_{p}u=\beta(u)| abla u| ^{p}+\lambda f(x)+\alpha, \] involves a source gradient term with natural growth, where $\beta$ is nonnegative, $\lambda>0,f(x)\geqq0$, and $\alpha$ is a nonnegative measure. The second one, of the form \[ -\Delta_{p}v=\lambda f(x)(1+g(v))^{p-1}+\mu, \] presents a source term of order $0, $where $g$ is nondecreasing, and $\mu$ is a nonnegative measure. Here $\beta$ and $g$ can present an asymptote. The correlation gives new results of existence, nonexistence, regularity and multiplicity of the solutions for the two problems, without or with measures. New informations on the extremal solutions are given when $g$ is superlinear.

Partial Differential Equations Correlation between two quasilinear elliptic problems with a source term involving the function or its gradient

Abstract: Thanks to a change of unknown we compare two elliptic quasilinear problems with Dirichlet data in a bounded domain of R N . The first one, of the form −� pu = β(u)|∇u| p + λf (x) ,w hereβ is nonnegative, involves a gradient term with natural growth. The second one, of the form −� pv = λf (x)(1 + g(v)) p−1 where g is nondecreasing, presents a source term of order 0. The correlation gives new results of existence, nonexistence and multiplicity for the two problems. To cite this article: H.A. Hamid, M.F. Bidaut-Veron, C. R. Acad. Sci. Paris, Ser. I ••• (••••).

Existence and Multiplicity for Elliptic Problems with Quadratic Growth in the Gradient

Abstract: We show that a class of divergence-form elliptic problems with quadratic growth in the gradient and non-coercive zero order terms are solvable, under essentially optimal hypotheses on the coefficients in the equation. In addition, we prove that the solutions are in general not unique. The case where the zero order term has the opposite sign was already intensively studied and the uniqueness is the rule.

NONLINEAR WEIGHTED p-LAPLACIAN ELLIPTIC INEQUALITIES WITH GRADIENT TERMS

Abstract: In this paper, we give sufficient conditions for the existence and nonexistence of nonnegative nontrivial entire weak solutions of p-Laplacian elliptic inequalities, with possibly singular weights and gradient terms, of the form div{g(|x|)|Du|p-2Du} ≥ h(|x|)f(u)l(|Du|). We achieve our conclusions by using a generalized version of the well-known Keller–Ossermann condition, first introduced in [2] for the generalized mean curvature case, and in [11, Sec. 4] for the nonweighted p-Laplacian equation. Several existence results are also proved in Secs. 2 and 3, from which we deduce simple criteria of independent interest stated in the Introduction.

Existence and regularity of positive solutions of elliptic equations of Schrödinger type

Abstract: We prove the existence of positive solutions with optimal local regularity of the homogeneous equation of Schrodinger type $$ - {\rm{div}}(A abla u) - \sigma u = 0{\rm{ in }}\Omega $$ for an arbitrary open Ω ⊆ ℝn under only a form-boundedness assumption on σ ∈ D′(Ω) and ellipticity assumption on A ∈ L∞(Ω)n×n. We demonstrate that there is a two-way correspondence between form boundedness and existence of positive solutions of this equation as well as weak solutions of the equation with quadratic nonlinearity in the gradient $$ - {\rm{div}}(A abla u) = (A abla v) \cdot abla v + \sigma {\rm{ in }}\Omega $$ As a consequence, we obtain necessary and sufficient conditions for both formboundedness (with a sharp upper form bound) and positivity of the quadratic form of the Schrodinger type operator H = −div(A∇·)-σ with arbitrary distributional potential σ ∈ D′(Ω), and give examples clarifying the relationship between these two properties.