Existence and uniqueness of the weak solution for aP-Laplacian problem

In this work we study the existence and uniqueness of weak solutions for a p-Laplacianproblem in RN of the form:−Δpu + m(x)|u|p−2u = f(x u(x)) (1)where 1 < p < N N 3 m 2 C(RNR) and 0 < m(x) < +1. Here f : RN × R ! R is a Carathéodory function that is decreasing with respect to the second variable.Compared to previous work we have replaced the Laplacian with the p-Laplacian.Using Monotone Operator Theory we establish the existence of a non-trivial solution.In RN the main difficulty is the lack of compactness. To overcome this we impose additional assumptions on the nonlinear term f(x u).