Problem with critical Sobolev exponent and with weight

Habib Yazidi Universite Paris 12, Val-de-Marne , France

Let Ω be a bounded domain in IR^{n} with n ≥ 3. We are concerned with the problem of existence of a function u satisfying the nonlinear elliptic equation:

−div(p(x)∇u) = u^{q−1} + λu in Ω,
u > 0  in Ω,
u = 0  on ∂Ω,

where p : Ω̄ ⟶ IR is a given positive weight such that p ∈ H^{1}(Ω) ∩ C(Ω̄), q = \frac{n+2}{n−2} is the critical exponent for the Sobolev embedding H_{0}^{1}(Ω) ⊂ L^{q}(Ω) and λ is a real constant. These results are obtained in collaboration with R. Hadiji.