Existence and multiplicity of solutions of quasilinear equations with convex or nonconvex reaction term

Abstract

We give existence, nonexistence and multiplicity results of nonnegative solutions for Dirichlet problems of the form $$ - {\Delta_p}v = \lambda f(x){\left( {1 + g(v)} \right)^{p - 1}}\quad {\text{in}}\ \Omega,\quad u = 0\quad {\text{on}}\ \partial \Omega, $$ where Δp is the p-Laplacian (p > 1), g is nondecreasing, superlinear, and possibly convex, λ > 0, and f ∈ L1 (Ω), f ≥ 0. New information on the extremal solutions is given. Equations with measure data are also considered.