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On a real Monge-Ampère functional. (English) Zbl 0724.35040
This paper is concerned with the Dirichlet problem for the Monge-Ampère equation \[ (1)\quad (\det \nabla^ 2u)^{1/n}=g(x,u,\nabla u)\text{ in } \Omega,\quad u=0\text{ on } \partial \Omega, \] where \(\Omega\) is a bounded convex domain in \(R^ n\), \(n\geq 2\). The point is, that this problem is studied without subjecting to monotonic condition or severe growth condition as in the cited references. Using the functional \[ J(u)=(-1/(n+1))\int_{\Omega} u\quad \det \nabla^ 2u\quad -\quad \int_{\Omega}G(x,u),\quad (where\quad G(x,z)=\int^{z}_{0}g^ n(x,s)ds), \] which has the property, that special critical points of J solve the above mentioned Dirichlet problem (1), and taking into account, that J is non-increasing along its logarithmic gradient flow, results on existence, non-existence, uniqueness, multiplicity of solutions of the problem (1) are presented as well as on properties of J (boundedness below, Dirichlet principle, mountain pass lemma), finally the results are compared with those on the ordinary Dirichlet problem for the Laplace equation.

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
35J20 Variational methods for second-order elliptic equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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