Global interval bifurcation and convex solutions for the Monge-Ampere equations

Abstract

In this article, we establish the global bifurcation result from the trivial solutions axis or from infinity for the Monge-Ampère equations with non-differentiable nonlinearity. By applying the above result, we shall determine the interval of γ, in which there exist radial solutions for the following Monge-Ampère equation det(D2u) = γα(x)F(-u), in B, u(x) = 0, on ∂B, where D2u = (∂2u/ ∂xi∂xj) is the Hessian matrix of u, where B is the unit open ball of ℝN, γ is a positive parameter. α ∈ C(B-, [0, +∞)) is a radially symmetric weighted function and α(r) := α(|x|) ≢ 0 on any subinterval of [0, 1] and the nonlinear term F ∈ C(ℝ+) but is not necessarily differentiable at the origin and infinity. We use global interval bifurcation techniques to prove our main results.