Solvability of a nonlinear elliptic problem involving $p-$Laplacian with Dirichlet condition

Abstract

In this work, we study a class of nonlinear elliptic equations of the form $Au=-\Delta _{p}u+ div \left( Vu\right) =R(x,u)$ in a bounded domain $\Omega \subset \mathbb{R}^n$, where $-\Delta _{p}u=- div \left( \left \vert \nabla u\right \vert ^{p-2}\nabla u\right)$ denotes the p-Laplacian operator, $V(x)$ is a given vector field, and $R(x)$ represents a source term. We establish the existence and uniqueness of weak solutions under suitable assumptions on $V$ and $R$. To establish the existence of weak solutions to the nonlinear elliptic problem we employ a fixed point approach based on the compactness of the associated operator. The existence result is obtained by proving that the composition $A^{-1}N_{R}$ is a compact and continuous mapping that admits at least one fixed point, where $N_{R}$ is the Nemyskii operator corresponding to a Caratheodory function $R$. This ensures the existence of a weak solution $u\in W_{0}^{1,p}\left( \Omega \right)$. For the uniqueness of solutions, we rely on the strict monotonicity of the p-Laplacian operator and appropriate growth and Lipschitz conditions on $R(x,u)$.