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:¿¿pu + m(x)|u|p¿2u = f(x, u(x)), (1)where 1 < p < N, N 3, m 2 C(RN,R), 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).