Abstract
We shall establish the global bifurcation results from the trivial solutions axis or from infinity for the Monge-Ampère equations: det(D2u)=λm(x)-uN+m(x)f1(x,-u,-u′,λ)+f2(x,-u,-u′,λ), 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, m∈C(B¯,[0,+∞)) is a radially symmetric weighted function and m(r):=m(x)≢0 on any subinterval of [0,1], λ is a positive parameter, and the nonlinear term f1,f2∈C(B¯×R+3,R+), but f1 is not necessarily differentiable at the origin and infinity with respect to u, where R+=[0,+∞). Some applications are given to the Monge-Ampère equations and we use global bifurcation techniques to prove our main results.
Highlights
The Monge-Ampere equations are a type of important fully nonlinear elliptic equations [1,2,3]
Kutev [9] and Delano [10] treated the existence of convex radial solutions of problem (1) with m ≡ 1, F = 0 and λm(−u)N + F = λ exp f(|x|, u, |∇u|), respectively
Following the above theory, we shall investigate the existence of radial solutions for the following problem: det (D2u) = γh (x) F (−u), in B, (5)
Summary
The Monge-Ampere equations are a type of important fully nonlinear elliptic equations [1,2,3]. Wang [13] and Hu and Wang [12] (m ≡ 0; F = f(−u)) considered the existence of strictly convex solutions for problem (2) by using fixed index theorem. Dai and Ma [15] and Dai [16] studied the Monge-Ampere equations (1) with λm(x)(−u)N + F equal λN((−u)N + g(−u)) and λNm(x)((−u)N + g(−u)), respectively, where g : [0, +∞) → [0, +∞) satisfies lims→0+ g(s)/sN = 0. Following the above theory (see Theorems 3 and 6), we shall investigate the existence of radial solutions for the following problem: det (D2u) = γh (x) F (−u) , in B, (5).
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