Nonhomogeneous Dirichlet Problems for the P-Laplacian

We study the existence, nonexistence and multiplicity of positive solutions for a family of problems -Δpu=fλ(x,u) in Ω,u=φon∂Ω, where λ> 0 is a parameter. The family we consider includes in particular the Pohozaev type equation -Δpu=λup∗-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 W1 , p and C1, a C1 , 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. © 2017, Springer-Verlag Berlin Heidelberg.