Abstract
First, the symmetry of classical solutions to the Monge–Ampère-type equations is obtained by the moving plane method. Then, the existence and nonexistence of radial solutions in a ball are got from the symmetry results. Finally, the existence and nonexistence of classical solutions to Hessian equations in bounded domains are considered.
Highlights
We firstly consider the Monge–Ampère-type equationsF (u) = det D2u + M(x, u, Du) = g(x, u, Du), (1.1)with D2u being the Hessian matrice of u, M(x, u, Du) being a given symmetric matrix function, and g being a positive function
The function u is called the potential function and it satisfies the optimal transportation equation det D2u – D2xθx, Y (x, Du) = η/| det Yp|, with the vector field Y : D × R × Rn → Rn, Y = Y (x, z, p) being independent of z, D being a domain in Rn, det Yp = 0, Y coming from a cost function θ : Rn × Rn → R, θ = θ(x, y) being determined by the equations θx x, Y (x, p) = p, and ηbeing a known nonnegative function η : D × R × Rn → R
Monge–Ampère-type equations have got a lot of interest [9, 10]
Summary
Zhang–Wang [13] have obtained that the classical solutions to the standard Monge– Ampère equations det D2u = e–u are symmetric. Consider the symmetry of solutions for the Dirichlet problem Theorem 1.2 (1) The Dirichlet problem (1.3) has no solution if 0 < κ < 1 and H is large enough.
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