On some differential equations involving a new kind of variable exponents

In this paper, we are concerned with some new first order differential equation defined on the whole real axis R . The principal part of the equation involves an operator with variable exponent p depending on the variable x ∈ R through the unknown solution while the nonlinear part involves the classical variable exponent p ( x ) . Such kind of situation is very related to the presence of the variable exponent and has not been treated before. Our existence result of nontrivial solution cannot be reached using standard variational or topological methods of nonlinear analysis and some sophisticated arguments have to be employed.