Positive solutions of the Dirichlet problem for the prescribed mean curvature equation
The authors consider the Dirichlet problem for the prescribed mean curvature equation \[ - \text{div}\big(\nabla u/ \sqrt{1+|\nabla u|^2}\big)= \lambda f(x,u) \quad\text{in }\Omega, \qquad u=0 \quad\text{on }\partial\Omega \tag \(*\) \] where \(\Omega\) is a bounded domain in \(\mathbb R^N\) (\(N\geq 2\)) with Lipschitz boundary, \(f:\Omega\times\mathbb R\rightarrow\mathbb R\) is a Carathéodory function, and \(\lambda\) is a real parameter. Solutions are taken in the bounded variation sense, namely, \(u\in \text{BV}(\Omega)\) is a solution of \((*)\) if \(f(\cdot,u)\in L^p(\Omega)\) for some \(p>N\) and \[ {\mathcal I}(v)-{\mathcal I}(u) \geq \lambda \int_\Omega f(x,u)(v-u)\,dx \] for every \(v\in \text{BV}(\Omega)\), where \({\mathcal I}:\text{BV}(\Omega)\rightarrow\mathbb R\) is defined by \[ {\mathcal I}(w) = \int_\Omega \sqrt{1+|\nabla w|^2} + \int_{\partial\Omega} | w_{|\partial\Omega}| \,d{\mathcal H}^{n-1}. \] In some cases stronger notions of solution are also considered. The authors are interested in conditions on \(F(x,s)=\int_0^s f(x,t)\,dt\) and on \(\lambda\) guaranteeing the existence and multiplicity of positive solutions of \((*)\). The proofs use variational methods and combine critical point theory, lower and upper solution methods, and elliptic regularization. The authors present an exhaustive list of results, including, as far as possible, all the possible configurations of limits of \(F(x,s)/s^2\) at zero and \(F(x,s)/s\) at infinity, which yield the existence of one, two, three or infinitely many positive solutions of \((*)\). Considering these limits at zero and infinity is suggested by the fact that the mean curvature operator behaves like the Laplacian near zero and like the 1-Laplacian \(\Delta_1u=\text{div}(\nabla u/|\nabla u|)\) near infinity.
scientific article; zbMATH DE number 5793318
scientific article