On Weak Solutions of a Mildly Nonlinear Dirichlet Problem
Michael B. Rosenzweig · SIAM Journal on Mathematical Analysis · 1971
This paper investigates minimal conditions on functions of the type $F(x,v(x))$ and on the boundary of a domain R to insure the existence and uniqueness of a weak solution to the mildly nonlinear Dirichlet problem \[ \begin{gathered} \Delta u(x) = F(x,u(x)),\quad x \in R, \hfill \\ u(x) = 0,\quad x \in \partial R, \hfill \\ \end{gathered} \] The principal technique is the construction of an operator by means of a linearization of the nonlinear equation. The operator is shown to satisfy the Schauder–Tikhonov theorem and the resulting fixed point is shown to be a solution of the mildly nonlinear problem.