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Abstract. We are concerned with the existence of solution for the Dirichlet problem x,u) lies in some sense between the first and the second eigenvalues of the p-Laplacian p . Extensions to more general operators which are (p − 1)-homogeneous at infinity are also considered.2000 Mathematics Subject Classification. 35J65.

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In this paper we study the regularity of the solutions to the problem ∆pu = |u| p−2 u in the bounded smooth domain Ω ⊂ R N , with |∇u| p−2 ∂u ∂ν = λV (x)|u| p−2 u+h as a nonlinear boundary condition, where ∂Ω is C 2,β with β ∈]0, 1[, and V is a weight in L s (∂Ω) and h ∈ L s (∂Ω) for some s ≥ 1. We prove that all solutions are in L ∞ (∂Ω) L ∞ (Ω), and using the D.Debenedetto's theorem of regularity in [1] we conclude that those solutions are in C 1,α Ω for some α ∈]0, 1[.

By applying the Ricceri's three critical points theorem, we show the existence of at least three solutions to the following elleptic problem:is a bounded domain of smooth boundary ∂Ω and ν is the outward normal vector on ∂Ω. p : Ω → R, a : Ω × R N → R N , f : Ω × R → R and g : ∂Ω × R → R are fulfilling appropriate conditions.