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Existence for some quasilinear elliptic systems with critical growth nonlinearity and L 1 data

Alaa N., Maach F., Mounir I.

Journal of Applied Analysis

2005

Vol. 11, nr 1

81-94

We prove the existence of weak solutions for some quasilinear elliptic reaction-diffusion systems with Dirichlet boundary conditions and satisfying to the two main properties: the positivity of the solutions and the balance law. The nonlinearity we consider here has critical growth with respect to the gradient and data are in L1.

Nonexistence of global solutions to a class of nonlinear differential inequalities and application to hyperbolic-elliptic problems

Alaa N., Guedda M.

Journal of Applied Analysis

2003

Vol. 9, nr 1

103-121

We consider the problem [wzór] posed in Ω x (0,+∞). Here Ω ⊂ Rn is a an open smooth bounded domain and φ is like [wzór] and ε = š1. We prove, in certain conditions on f and φ that there is absence of global solutions. The method of proof relies on a simple analysis of the ordinary inequality of the type w'' + δw' ≥ αw + βwp. It is also shown that a global positive solution, when it exists, must decay at least exponentially.

Convergence of Toland's critical points for sequences of D.C. functions and application to the resolution of semilinear elliptic problems

Yassine A., Alaa N., Elhilali Alaoui A.

Control and Cybernetics

2001

Vol. 30, no 4

405-417

We prove that if a sequence (fn)n of D.C. functions (Difference of two Convex functions) converges to a D.C. function f in some appropriate way and if un is a critical point of fn, in the sense described by Toland, and is such that (un)n converges to u, then a is a critical point of f, still in Toland's sense. We also build a new algorithm which searches for this critical point u and then apply it in order to compute the solution of a semilinear elliptic equation.