The Neumann problem for quasilinear differential equations

Tiziana Cardinali, Nikolaos S. Papageorgiou and Raffaella Servadei

Address.University of Perugia, Department of Mathematics and Computer Science, via Vanvitelli 1, Perugia 06123, Italy

National Technical University, Department of Mathematics, Zografou Campus, Athens 15780, Greece

University of Roma `Tor Vergata', Department of Mathematics, via della Ricerca Scientifica, Roma 00133, Italy


Abstract. In this note we prove the existence of extremal solutions of the quasilinear Neumann problem $-(|x'(t)|^{p-2}x'(t))' = f(t, x(t), x'(t))$, a.e. on $T$, $x'(0) = x'(b) =0$, $2\leq p < \infty$ in the order interval $[\psi,\varphi]$, where $\psi$ and $\varphi$ are respectively a lower and an upper solution of the Neumann problem.

AMSclassification. 35J60, 35J65.

Keywords.  Upper solution, lower solution, order interval, truncation function, penalty function, pseudomonotone operator, coercive operator, Leray-Schauder principle, maximal solution, minimal solution.