Nonhomogeneous Dirichlet problems for the p-Laplacian

We study the existence, nonexistence and multiplicity of positive solutions for a family of problems - Δ p u = f λ ( x , u ) in Ω , u = φ on ∂ Ω , where λ > 0 is a parameter. The family we consider includes in particular the Pohozaev type equation - Δ p u = λ u p ∗ - 1 . The main new feature is the consideration of the p-Laplacian - Δ p together with a nonzero boundary condition φ . In order to deal with these nonhomogeneous problems, it has been important to extend to this new context several basic results such as the Brezis-Nirenberg theorem on local minimization in W 1 , p and C 1 , a C 1 , α estimate for a family of equations with critical growth, and a variational approach to the method of upper–lower solutions. These extensions have an independent interest for applications in other situations.