Structural Stability of Nonlinear Elliptic p(u)-Laplacian Problem with Robin Type Boundary Condition

Abstract

We study in this chapter the following nonlinear elliptic boundary value problem,

$$\displaystyle b(u)-\mbox{div}a(x,u,\nabla u)=f\qquad \qquad \mbox{in}\qquad \qquad \Omega ,\qquad \qquad a(x,u,\nabla u).\eta =-|u|{ }^{r(x,u)-2}u\qquad \qquad \mbox{on}\qquad \qquad \partial \Omega , $$

where Ω is a bounded open domain in $\mathbb {R}^{N}$, N ≥ 3, with smooth boundary ∂ Ω. We prove the existence and uniqueness of weak solution for f ∈ L¹( Ω) and structural stability result.