Multiplicity of Weak Positive Solutions for Fractional & Laplacian Problem with Singular Nonlinearity

where Ω ⊂ R (푁 ≥ 3) is an open bounded domain with smooth boundary 휕Ω , 푁 > 2푠(0 < 푠 < 1), 1 ≤ 푞 < 푝 < 푁/푠, 0 < 훾 < 1 < 훽 < 푝∗ 푠 − 1 , is a positive parameter. e weight functions 푎 : Ω → R is in 푝∗ 푠 /(푝∗ 푠 +훾−1) with 푎(푥) > 0 for almost every 푥 ∈ Ω, 푏 : Ω → R is bounded with 푏(푥) > 0 for almost every 푥 ∈ Ω, and 푝∗ 푠 = 푝푁/(푁 − 푝푠) denotes the critical Sobolev exponent. (−Δ) , with 푟 ∈ {푝, 푞}, is the fractional -Laplacian operator defined for any 푢 ∈ C∞ 푐 (R푁) by

In recent years, the fractional Laplacian problems have been extensively investigated. For more details, we cite the reader to [11][12][13][14][15]. ere are many different definitions of weak solutions for the fractional Laplacian equation (3). In [16], Fang say that 푢 ∈ 퐻 푠 0 (Ω) is a weak solution of (3) with 휆 = 0 and 푎 ≡ 1 if the identity holds. In virtue of the method of sub-supersolution, the author gives the sufficient conditions for the existence and uniqueness of positive solution.
Very recently, great attention has been devoted to the study of fractional -Laplacian problems, see for instance [17][18][19][20]. However, in literature, there are only a few papers [21][22][23] dealing with fractional & problems. Motivated by the works [1,23,24], in this paper, we investigate the existence and multiplicity of solutions for the fractional & Laplacian problem (1) and extend the main results of Wang and Zhang [1]. (1) is article is organized as follows. In Section 2, we give some notations and preliminaries. Section 3 is devoted to proving that problem (1) has at least two positive solutions for sufficiently small.
Proof. By using the definition of N (Ω) and (16), we have is implies that J (Ω) is coercive and bounded from below on N (Ω). e proof is complete. ☐

Journal of Function Spaces 4 which yields that
Passing to the limit as 푛 → ∞, we get J = .
In the following, we will show that 푢 휆 ∈ N + 휆 (Ω). It suffices to prove that → strongly in 푊 푠,푝 0 (Ω). By J = and we have . e proof is complete. ☐

Main Result
Similar to the proof of Lemma 3.7 in [1], we can easily obtain the following Lemma.

Conclusions
In this paper, the existence and multiplicity of positive solutions for a class of fractional & Laplacian problem with singular nonlinearity have been investigated. It is worthy to point out that few studies have been done on this issue. By means of the variational method, Nehari manifold method and some analysis techniques, the sufficient conditions of existence and multiplicity of positive solutions to this problem have been presented in eorem 13. Our results generalize the main conclusions of Wang and Zhang in [1].

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest.