EQUATIONS ET DES SYSTEMES ELLIPTIQUES SEMI-LINEAIRES

The Nehari Manifold

Nehari has introduced a variational method very useful in critical point theory and eventually came to bear his name. He considered a boundary value problem for a certain nonlinear second-order ordinary differential equation in an interval [a; b] and proved that it has a nontrivial solution which may be obtained by constrained minimization. To describe Nehari’s method in an abstract setting, let E be a Banach space and J 2 C 1 (E; R) a functional. The Frechet derivative of J at u, J 0 (u); is an element of the dual space E 0 . Suppose u 6= 0 is a critical point of J, i.e., J 0 (u) = 0: Then necessarily u is contained in the set N = n u 2 En f0g : D J 0 (u); uE = 0o : 30 Chapter 2. Nehari manifold and Fibering method So N is a natural constraint for the problem of finding nontrivial critical points of J(u) by minimizing the energy functional J on the constraint N is called the Nehari manifold. Set c := inf J(u): u2N Under appropriate conditions on J one hopes that c is attained at some u0 2 N and that u0 is a critical poin

Fibering method

 At the end of the 1990s, the fibering method or the decomposition method introduced by Pohozaev for investigating some variational problems, and its applications to nonlinear elliptic equations. Let X and Y be Banach spaces, and let A be an nonlinear operator acting from X to Y . We consider the equation (2. 5. 1) A(u) = h: The fibering method is based on representation of solutions of equation (2. 5. 1) in the form u = tu: Where t is a real parameter, t 6= 0 in some open J  R: Now, we give a complete description of the fibering method, we begin by defining the fibre map of the 31 Chapter 2. Nehari manifold and Fibering method following (t) : R + ! R such that (t) = J(tu); then, we calculate  0 (t); 00(t) the first and second derivative of (t). We decompose N into three parts N +; N 

Ekeland’s variational principle 

In general, it is not clear that a bounded and lower semi-continuous functional E actually attains its infimum. The analytic function f(x) = arctan x, for example, neither attains its infimum nor its supremum on the real line. A variant due to Ekeland of Dirichlet’s principle, however, permits one to construct minimizing sequences for such functionals E whose elements um each minimize a functional Em, for a sequence of functionals fEmg converging locally uniformly to E: Theorem 2.6.1. [31] Let E be a reflexive Banach space with norm k:k, and J : E ! R is coercive and weakly lower semi-continuous on E , that is, suppose the following conditions are fullfilled:  J(u; v) ! 1 as k(u; v)k ! 1; (u; v) 2 E:  For any (u; v) 2 E; any sequence (un; vn) in E such that (un; vn) + (u; v) weakly in E there holds J(u; v)  lim inf n!1 J(un; vn): 38 Chapter 2. Nehari manifold and Fibering method  Then J is bounded from below on E and attains its infimum in E such that J(u0; v0) = inf E J: Theorem 2.6.2. [31] Let M be a complete metric space with metric d, and let J : M ! R[f+1g be lower semi-continuous, bounded from below, and 6= 1:Then for any ;  > 0; any u 2 M with J(u)  inf M J(u) + ; there is an element v 2 M strictly minimizing the functional Jv(w)  J(w) +   d(v; w): Moreover, we have J(v)  J(u); d(u; v)  :

Singular fractional elliptic system

In this chapter, we apply the Nehari Manifold, Fibering method, and Ekeland’s variational principle to establish the existence and multiplicity results of nontrivial positive solutions for the system (p): We consider the following singular fractional elliptic system (P.