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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 $$L^{\infty }(\Omega )\cap BV(\Omega )$$.

##### MSC:
 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