Multiplicity of solutions of Kirchhoff-type fractional Laplacian problems with critical and singular nonlinearities

Duan, Qingwei and Guo, Lifeng and Zhang, Binlin (2023) Multiplicity of solutions of Kirchhoff-type fractional Laplacian problems with critical and singular nonlinearities. ELECTRONIC JOURNAL OF QUALITATIVE THEORY OF DIFFERENTIAL EQUATIONS, 2023 (45). pp. 1-28. ISSN 1417-3875

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Abstract

In this article, the following Kirchhoff-type fractional Laplacian problem with singular and critical nonlinearities is studied: a +b∥u∥2µ−2 (−∆)su = λl(x)u2∗ s−1 +h(x)u−γ, in Ω, u >0, in Ω, u =0, in RN\Ω, where s ∈ (0,1), N > 2s, (−∆)s is the fractional Laplace operator, 2∗ s = 2N/(N −2s) is the critical Sobolev exponent, Ω ⊂ RN is a smooth bounded domain, l ∈ L∞(Ω) 2∗ s is a non-negative function and max{l(x),0} ̸ ≡ 0, h ∈ L 2∗ s+γ−1 (Ω) is positive almost everywhere in Ω, γ ∈ (0,1), a > 0,b > 0, µ ∈ [1,2∗ s/2) and parameter λ is a positive constant. Here we utilize a special method to recover the lack of compactness due to the appearance of the critical exponent. By imposing appropriate constraint on λ, we obtain two positive solutions to the above problem based on the Ekeland variational principle and Nehari manifold technique.

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