Existence and multiplicity of positive solutions for a critical fractional Laplacian equation with singular nonlinearity
Fractional Calculus and Applied Analysis - Trang 1-27 - 2024
Tóm tắt
In this paper, we consider the following problem
$$\begin{aligned} {\left\{ \begin{array}{ll} (-\varDelta )^{s} u=g(x) u^{2_{s}^{*}-1}+\lambda u^{-\gamma }, &{} \text { in } \varOmega , \\ u>0, \text { in } \varOmega , \quad u=0, &{} \text { on } \partial \varOmega , \end{array}\right. } \end{aligned}$$
where
$$\varOmega \subset {\mathbb {R}}^{N}(N > 2s)$$
is a smooth bounded domain,
$$s\in (0,1)$$
,
$$\lambda $$
is a positive constant,
$$0<\gamma <1$$
,
$$2_{s}^{*}=\frac{2 N}{N-2s}$$
and
$$(-\varDelta )^{s} $$
is the spectral fractional Laplacian. Based upon the Nehari manifold and using variational method we relate the number of positive solutions to the global maximum of the coefficient of the critical nonlinearity g.
Tài liệu tham khảo
Abatangelo, N., Valdinoci, E.: Getting Acquainted with the Fractional Laplacian. Springer INdAM Series (2019)
Anello, G., Faraci, F.: Two solutions for a singular elliptic problem indefinite in sign. NoDEA Nonlinear Differ. Equ. Appl. 22, 1429–1443 (2015)
Barrios, B., De Bonis, I., Medina, M., Peral, I.: Semilinear problems for the fractional Laplacian with a singular nonlinearity. Open Math. 13, 390–407 (2015)
Brandle, C., Colorado, E., de Pablo, A.: A concave-convex elliptic problem involving the fractional Laplacian. Proc. R. Soc. Edinb. Sect. A 143, 39–71 (2013)
Brézis, H., Lieb, E.: A relation between pointwise convergence of functions and convergence of functionals. Proc. Am. Math. Soc. 88(3), 486–490 (1983)
Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Comm. Partial Differ. Equ. 32, 1245–1260 (2007)
Capella, A., Dávila, J., Dupaigne, L., Sire, Y.: Regularity of radial extremal solutions for some non-Local semilinear equations. Commun. Partial Differ. Equ. 36, 1353–1384 (2011)
Crandall, M.G., Rabinowitz, P.H., Tartar, L.: On a Dirichlet problem with a singular nonlinearity. Commun. Partial Differ. Equ. 2, 193–222 (1977)
Echarghaoui, R., Khouakhi, M., Masmodi, M.: Existence and multiplicity of positive solutions for a class of critical fractional Laplacian systems. J Elliptic Parabol. Equ. 8, 813–835 (2022). https://doi.org/10.1007/s41808-022-00177-6
Echarghaoui, R., Masmodi, M.: Two disjoint and infinite sets of solutions for a concave-convex critical fractional Laplacian equation. Fract. Calc. Appl. Anal. 25, 1604–1629 (2022). https://doi.org/10.1007/s13540-022-00060-0
Giacomoni, J., Mukherjee, T., Sreenadh, K.: Positive solutions of fractional elliptic equation with critical and singular nonlinearity. Adv. Nonlinear Anal. 6, 327–354 (2017)
Ghanmi, A., Saoudi, K.: The Nehari manifold for a singular elliptic equation involving the fractional Laplace operator. Fract. Differ. Calc. 6(2), 201–217 (2016)
Mukherjee, T., Sreenadh, K.: Fractional elliptic equations with critical growth and singular nonlinearities. Electron. J. Differ. Equ. 54, 1–23 (2016)
Saoudi, K.: A critical fractional elliptic equation with singular nonlinearities. Fract. Calc. Appl. Anal. 20(6), 1507–1530 (2017). https://doi.org/10.1515/fca-2017-0079
Saoudi, K., Ghosh, S., Choudhuri, D.: Multiplicity and Hölder regularity of solutions for a nonlocal elliptic PDE involving singularity. J. Math. Phys. 60, 101509 (2019)
Servadei, R., Valdinoci, E.: The Brezis-Nirenberg result for the fractional Laplacian. Trans. Am. Math. Soc. 367, 67–102 (2015)
Servadei, R., Valdinoci, E.: On the spectrum of two different fractional operators. Proc. R. Soc. Edinburgh Sect. A 144, 831–855 (2014)
Stinga, P., Torrea, J.: Extension problem and Harnack’s inequality for some fractional operators. Commun. Partial Differ. Equ. 35, 2092–2122 (2010)
Sun, Y.: Compatibility phenomena in singular problems. Proc. R. Soc. Edinb. Sect. A 143, 1321–1330 (2013)
Sun, Y., Wu, S.: An exact estimate result for a class of singular equations with critical exponents. J. Funct. Anal. 260, 1257–1284 (2011)
Tai, D., Fang, Y.: Existence and uniqueness of positive solutions to fractional Laplacians with singular nonlinearities. Applied Mathematics Letters 119, 107227 (2021)
Wang, Q.F.: The Nehari manifold for a fractional Laplacian equation involving critical nonlinearities. Commun. Pure Appl. Anal. 6, 2261–2281 (2018)