Superposition operators between Sobolev spaces and a non-existence result of higher-order regular solutions for the [formula omitted]-Laplacian

Abstract

In this paper, a well-known non-existence result of Pohožaev for a Dirichlet problem with Laplacian is generalized to a non-existence result of higher-order regular strong solutions for a Dirichlet problem with p-Laplacian subjected to the following natural principle: the stronger (respectively weaker) are the assumptions on the given data, the larger (respectively smaller) is the Sobolev space in which no nontrivial solutions can be found. To do this, we use some recent developments on superposition operators between higher-order Sobolev spaces.