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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.

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
Full Text: DOI EuDML
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