Positive, Extremal and Nodal Solutions for Nonlinear Parametric Problems

We consider a nonlinear parametric problem driven by the

-Laplace differential operator. For all large enough values of the parameter

\lambda λ

, we show that the problem has a smallest positive solution

u_{\lambda}^*\in C^1_0 (\overline{\Omega}) u λ ∗ ∈ C 0 1 ( Ω ‾ )

. We examine the monotonicity and continuity properties of the map

\lambda\longmapsto u^*_{\lambda} λ ⟼ u λ ∗

. Finally we establish the existence of nodal (sign changing) solutions.