Principal Eigenvalue of p-Laplacian Operator in Exterior Domain

Abstract

We consider an eigenvalue problem of the form

$$\left.\begin{array}{cl}-\Delta_{p} u = \lambda\, K(x)|u|^{p-2}u \quad \mbox{in}\quad \Omega^e\\ u(x) =0 \quad \mbox{for}\quad \partial \Omega\\ u(x) \to 0 \quad \mbox{as}\quad |x| \to \infty,\end{array} \right \}$$

where ${\Omega \subset \mathrm{I\!R\!}^N}$ is a simply connected bounded domain, containing the origin, with C ² boundary ${\partial \Omega}$ and ${\Omega^e:=\mathrm{I\!R\!^N} \setminus \overline{\Omega}}$ is the exterior domain, ${1 < p < N, \Delta_{p}u:={\rm div}(|\nabla u|^{p-2} \nabla u)}$ is the p-Laplacian operator and ${K \in L^{\infty}(\Omega^e) \cap L^{N/p}(\Omega^e)}$ is a positive function. Existence and properties of principal eigenvalue λ ₁ and its corresponding eigenfunction are established which are generally known in bounded domain or in ${\mathrm{I\!R\!}^N}$. We also establish the decay rate of positive eigenfunction as ${|x| \to \infty}$ as well as near ∂Ω.