Non-homogeneous problem for fractional Laplacian involving critical Sobolev exponent




Cheng, Kun
Wang, Li

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Texas State University, Department of Mathematics


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.



non-homogeneous, fractional Laplacian, critical Sobolev exponent, variational method


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.


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