Study on Existence of Solutions for p-Kirchhoff Elliptic Equation in ℝ^N with Vanishing Potential

Abstract

In this paper, we study the existence of positive solutions to p−Kirchhoff elliptic problem

$\begin{array}{@{}rcl@{}} \left\{\begin{array}{lllllll} &\left(a+\mu\left({\int}_{\mathbb{R}^{N}}\!(|\nabla u|^{p}+V(x)|u|^{p})dx\right)^{\tau}\right)\left(-{\Delta}_{p}u+V(x)|u|^{p-2}u\right)=f(x,u), \quad \text{in}\; \mathbb{R}^{N}, \\ &u(x)>0, \;\;\text{in}\;\; \mathbb{R}^{N},\;\; u\in \mathcal{D}^{1,p}(\mathbb{R}^{N}), \end{array}\right.\!\!\!\! \\ \end{array}$     (0.1)

where a, μ > 0, τ > 0, and f(x, u) = h ₁(x)|u|^m−2 u + λ h ₂(x)|u|^r−2 u with the parameter λ ∈ ℝ, 1 < p < N, 1 < r < m < $p^{*}=\frac {pN}{N-p}$, and the functions h ₁ (x), h ₂(x) ∈ C(ℝ^N) satisfy some conditions. The potential V(x) > 0 is continuous in ℝ^N and V(x)→0 as |x|→+∞. The nontrivial solution forb Eq. (1.1) will be obtained by the Nehari manifold and fibering maps methods and Mountain Pass Theorem.