Existence and regularity results for p-Laplacian boundary value problems (Q489975)

The main purpose of the paper is to introduce some of the main tools to study nonlinear boundary value problems. In particular, the Dirichlet problem for the \(p\)-Laplace operator is concerned, which is the simplest example. The paper is divided into four sections. After introductory section, the second one deals with existence and regularity results when the right hand side belongs to the dual space of \(W^{1,p}(D)\). In this case, the model problem (P) is a variational boundary value problem. In Sections 3 and 4 the problem (P) is considered, when the right hand side is a function which does not belong to the dual space \(W_{0}^{1,p}(D)\). Next, the existence of distributional solutions belonging to a function space strictly contained in \(W_{0}^{1,1}(D)\) is shown. On the other hand, it is proved the existence of solutions belonging to \(W_{0}^{1,1}(D)\) and not belonging to \(W_{0}^{1,q}(D ), 1<q<p\). It is pointed out that existence results of \(W_{0}^{1,1}(D)\) distributional solutions is not so usual in elliptic problems.