Non-homogeneous problem for fractional Laplacian involving critical Sobolev exponent
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Date
2017-12-11
Authors
Cheng, Kun
Wang, Li
Journal Title
Journal ISSN
Volume Title
Publisher
Texas State University, Department of Mathematics
Abstract
In this article, we study the existence of positive solutions for the nonhomogeneous fractional equation involving critical Sobolev exponent
(-∆)su + λu = up + μƒ(x), u > 0 in Ω,
u = 0, in ℝN \ Ω,
where Ω ⊂ ℝN is a smooth bounded domain, N ≥ 1, 0 < 2s < min{N, 2}, λ and μ > 0 are two parameters, p = N+2s/N-2s and ƒ ∈ C0,α(Ω̅), where α ∈ (0, 1). ƒ ≥ 0 and ƒ ≢ 0 in Ω. For some λ and N, by the barrier method and mountain pass lemma, we prove that there exists 0 < μ̅ ≔ μ̅, (s, μ, N) < +∞ such that there are exactly two positive solutions if μ ∈ (0, μ̅) and no positive solutions for μ > μ̅ . Moreover, if μ = μ̅, there is a unique solution (μ̅; uμ̅), which means that (μ̅/ uμ̅) is a turning point for the above problem. Furthermore, in case λ > 0 and N ≥ 6s if Ω is a ball in ℝN and ƒ satisfies some additional conditions, then a uniqueness existence result is obtained for μ > 0 small enough.
Description
Keywords
Non-homogeneous, Fractional Laplacian, Critical Sobolev exponent, Variational method
Citation
Cheng, K., & Wang, L. (2017). Non-homogeneous problem for fractional Laplacian involving critical Sobolev exponent. <i>Electronic Journal of Differential Equations, 2017</i>(304), pp. 1-24.