Positive, Extremal and Nodal Solutions for Nonlinear Parametric Problems

We consider a nonlinear parametric problem driven by the $p$-Laplace differential operator. For all large enough values of the parameter $\lambda$, we show that the problem has a smallest positive solution $u_{\lambda}^*\in C^1_0 (\overline{\Omega})$. We examine the monotonicity and continuity properties of the map $\lambda\longmapsto u^*_{\lambda}$. Finally we establish the existence of nodal (sign changing) solutions.