On some equation defined on the whole euclidean space ℝN and involving the p(u)‐Laplacian

Abstract

In this paper, we deal with some new kinds of problems defined on the whole euclidean space involving an operator with variable exponent which depends on the unknown solution u. In the first part of this work, we treat a local second‐order partial differential equation, that is, when the exponent depends on the variable through the unknown solution u. In the second part, we study the nonlocal version of the first problem; more precisely, we are interested in the situation where the exponent depends on a scalar function of u. A suitable approximation scheme is performed, and the process of passage to the limit is completed using some sophisticated arguments. These results are immediately extended to some fourth‐order problem with variable exponents depending on the unknown function u as well as one or many of its partial derivatives ∂u/∂xj, 1 ≤ j ≤ N.