Elliptic equations with diffusion parameterized by the range of nonlocal interactions

Armel Andami Ovono (1) , Arnaud Rougirel (1)

1 LMA [Poitiers] - Laboratoire de mathématiques et applications [UMR 6086]

We consider quasilinear elliptic equations where the diffusion at each point depends on all the values of the solution in a neighborhood of this point. The size of this neighborhood is parameterized by some non negative number which represents the range of nonlocal interactions. For fixed values of the parameter, the issue of the existence and local uniqueness of the \so is addressed. In a radial symmetric setting, we give pointwise estimates of the \sos and prove the existence of multiple solutions. Regarding bifurcation theory, we show that many local branches of \sos may exist while, among them, only one is global and has no bifurcation point.

Mots clés: Nonlocal elliptic equations; branches of solutions; parameter; nonlocal diffusion

Domaines: Equations aux dérivées partielles [math.AP]

Armel Andami Ovono, Arnaud Rougirel. Elliptic equations with diffusion parameterized by the range of nonlocal interactions. 2008. ⟨hal-00279310v2⟩