Existence of multiple solutions of p-fractional Laplace operator with sign-changing weight function
Existence of multiple solutions of p-fractional Laplace operator with sign-changing weight function
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DOI：
10.1515/anona-2014-0017
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发表时间：
2014-08
影响因子：
4.2
通讯作者：
Sarika Goyal;K. Sreenadh
中科院分区：
文献类型：
--
作者：
Sarika Goyal;K. Sreenadh
Abstract In this article, we study the following p-fractional Laplacian equation: (P λ )-2∫ ℝ n |u(y)-u(x)| p-2 (u(y)-u(x)) |x-y| n+pα dy=λ|u(x)| p-2 u(x)+b(x)|u(x)| β-2 u(x)inΩ,u=0inℝ n ∖Ω,u∈W α,p (ℝ n ),$ (P_{\lambda }) \quad -2\int _{\mathbb {R}^n}\frac{|u(y)-u(x)|^{p-2}(u(y)-u(x))}{|x-y|^{n+p\alpha }} dy = \lambda |u(x)|^{p-2} u(x) + b(x)|u(x)|^{\beta -2}u(x) \quad \text{in } \Omega , \quad u = 0 \quad \text{in }\mathbb {R}^n \setminus \Omega ,\, u\in W^{\alpha ,p}(\mathbb {R}^n), $ where Ω is a bounded domain in ℝn with smooth boundary, n > pα, p ≥ 2, α ∈ (0,1), λ > 0 and b : Ω ⊂ ℝn → ℝ is a sign-changing continuous function. We show the existence and multiplicity of non-negative solutions of (Pλ) with respect to the parameter λ, which changes according to whether 1 < β < p or p < β < p* with p* = np(n-pα)-1 respectively. We discuss both cases separately. Non-existence results are also obtained.
Abstract In this article, we study the following p-fractional Laplacian equation: (P λ )-2∫ ℝ n |u(y)-u(x)| p-2 (u(y)-u(x)) |x-y| n+pα dy=λ|u(x)| p-2 u(x)+b(x)|u(x)| β-2 u(x)inΩ,u=0inℝ n ∖Ω,u∈W α,p (ℝ n ),$ (P_{\lambda }) \quad -2\int _{\mathbb {R}^n}\frac{|u(y)-u(x)|^{p-2}(u(y)-u(x))}{|x-y|^{n+p\alpha }} dy = \lambda |u(x)|^{p-2} u(x) + b(x)|u(x)|^{\beta -2}u(x) \quad \text{in } \Omega , \quad u = 0 \quad \text{in }\mathbb {R}^n \setminus \Omega ,\, u\in W^{\alpha ,p}(\mathbb {R}^n), $ where Ω is a bounded domain in ℝn with smooth boundary, n > pα, p ≥ 2, α ∈ (0,1), λ > 0 and b : Ω ⊂ ℝn → ℝ is a sign-changing continuous function. We show the existence and multiplicity of non-negative solutions of (Pλ) with respect to the parameter λ, which changes according to whether 1 < β < p or p < β < p* with p* = np(n-pα)-1 respectively. We discuss both cases separately. Non-existence results are also obtained.