On the existence of solutions of the Dirichlet problem for nonlinear elliptic equations
The author extends the method of super and sub solutions to the Dirichlet problem \[ (*)\quad D_ i(a_{ij}(x,u)D_ ju)+b(x,u,Du)=0\text{ in } \Omega,\quad u=f\text{ on } \partial \Omega \] in a bounded domain \(\Omega \subset {\mathbb{R}}^ n\). It is assumed that \(\partial \Omega \subset {\mathbb{C}}^ 2\), \(f\in L^{\infty}(\partial \Omega)\), and \[ | b(x,u,p)| \leq C(| u|)(1+| p|^ 2)\text{ for all } (x,u,p)\in \Omega \times {\mathbb{R}}\times {\mathbb{R}}^ n, \] \({\mathbb{C}}\) is an increasing function of its argument. The conditions on \(a_{ij}\) are: \[ | a_{ij}(x,u)| \leq A(1+| u|),\quad | D_ ka_{ij}(x,u)| \leq A(1+| u|), \] k,j,i\(=1,...,n\), \(A>0\), a constant, and \(a_{ij}\) is uniformly elliptic for (x,u)\(\in \Omega \times {\mathbb{R}}\). He then proves that (*) has a solution in an appropriate Sobolev space.
scientific article; zbMATH DE number 4148698
scientific article