Remarks on variational methods and lower-upper solutions
The authors consider the following class of equations \[ \begin{cases} -\Delta u(x)= f\bigl(x,u(x) \bigr),\quad x\in\Omega,\\ u=0\text{ on }\partial \Omega,\end{cases} \tag{1} \] where \(\Omega\subset\mathbb{R}^N\) is a bounded smooth domain. Letting \(F(x,t)=\int^t_0 f(x,s)ds\), the energy functional associated to (1) is defined by the formula \[ J(u)=\int_\Omega \Bigl(\textstyle {1\over 2}|\nabla u|^2-F(x,u) \Bigr)dx,\quad u\in \overset\circ W^1_2 (\Omega). \] Under suitable structural assumptions imposed on \(f,J\) turns out to be of class \(C^1\), and weak solutions of (1) are sought as critical points of \(J\). Assuming the existence of lower and upper solutions with appropriate ordering relations, the authors' exploit the variational structure of problem (1) to show the existence of local \(J\)-minimizers.
scientific article; zbMATH DE number 1406550