Differential and Integral Equations Volume 25, Numbers 7-8 (2012) , 657–664 QUASILINEAR EQUATIONS INVOLVING NONLINEAR NEUMANN BOUNDARY CONDITIONS Leonelo Iturriaga 1 Departamento de Matem´ atica, Universidad T´ ecnica Federico Santa Mar´ ıa Avenida Espa˜ na 1680, Casilla 110-V, Valpara´ ıso, Chile Sebasti´ an Lorca Instituto de Alta Investigaci´ on, Universidad de Tarapac´ a Casilla 7D, Arica, Chile Eugenio Saavedra and Pedro Ubilla 2 Departamento de Matem´ aticas y C. C., Universidad de Santiago de Chile Casilla 307, Correo 2, Santiago, Chile (Submitted by: Jesus Ildefonso Diaz) Abstract. We study the multiplicity of positive solutions of the problem -Δpu + |u| p-2 u =0 in a bounded smooth domain Ω ⊂ R N , with a nonlinear boundary con- dition given by |∇u| p-2 ∂u/∂ν = λf (u)+ μϕ(x)|u| q-1 u, where f is continuous and satisfies some kind of p-superlinear condition at 0 and p-sublinear condition at infinity, 0 <q<p - 1 and ϕ is L β (∂Ω) for some β> 1. In addition, we consider the case q = 0, where the nonlinear boundary condition becomes an elliptic inclusion. Our approach allows us to show that these problems have at least six nontrivial solutions, three positive and three negative, for some positive parameters λ and μ. The proof is based on variational arguments. 1. Introduction and statement of the results Consider the following class of nonlinear problems: (P ) λ,μ ( -Δ p u + |u| p-2 u =0 in Ω, |∇u| p-2 ∂u ∂ν = λf (u)+ μϕ(x)|u| q-1 u on ∂ Ω. Accepted for publication: November 2011. AMS Subject Classifications: 35J60, 35J25, 35J70. 1 Partially supported by FONDECYT grant N o 11080203 2 Partially supported by FONDECYT grant N o 1080430. 657