Existence and multiplicity results for the prescribed mean curvature equation via lower and upper solutions. (English) Zbl 1240.35131

The paper studies a mixed problem for the prescribed mean curvature equation \(-div (\nabla u/\sqrt {1+| \nabla u| ^2})=f(x,u)\) in \(\Omega \), \(u=0\) on \(\Gamma _D \), \(\partial u/\partial \nu =0\) on \(\Gamma _N \). Here \(\Omega \) is a bounded domain with Lipschitz boundary, \(\Gamma _D \) is an open subset of \(\partial \Omega \) and \(\Gamma _N =\partial \Omega \setminus \Gamma _D \), the function \(f:\Omega \times R\to R\) satisfies the Carathéodory conditions. Under certain conditions on \(f\) it is proved the existence of a solution \(u\in L^{\infty }(\Omega )\cap BV(\Omega )\). The proof is based on variational techniques and a lower and upper solutions method specially developed for this problem. The regularity of solutions of the problem is also studied. At the end the uniqueness of a solution of the problem is discussed. Conditions on \(f\) and \(\Omega \) are established under which there exist infinitely many solutions from \(L^{\infty }(\Omega )\cap BV(\Omega )\).


35J25 Boundary value problems for second-order elliptic equations
35J20 Variational methods for second-order elliptic equations
35J60 Nonlinear elliptic equations
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
49Q20 Variational problems in a geometric measure-theoretic setting