Remarks on variational methods and lower-upper solutions. (English) Zbl 0941.35031

Conti, Monica; Merizzi, Luca; Terracini, Susanna

NoDEA, Nonlinear Differ. Equ. Appl. 6, No. 4, 371-393 (1999).

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.

Reviewer: M.Fuchs (Saarbrücken)

MSC:

35J65	Nonlinear boundary value problems for linear elliptic equations
35B38	Critical points of functionals in context of PDEs (e.g., energy functionals)
35A15	Variational methods applied to PDEs

Keywords:

semilinear elliptic equations; existence and multiplicity of solutions; lower and upper solutions