No abstract

In this paper, we deal with some new kinds of problems defined on the whole euclidean space R N , N ≥ 1, 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 x ∈ R N 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/𝜕x j , 1 ≤ j ≤ N.

No abstract