A Global multiplicity result for a very singular critical nonlocal equation

Jacques Giacomoni, Tuhina Mukherjee, Konijeti Sreenadh

DOI: http://dx.doi.org/10.12775/TMNA.2019.049


In this article we show the global multiplicity result for the following nonlocal singular problem \begin{equation*} (-\Delta)^s u = u^{-q} + \lambda u^{{2^*_s}-1}, \quad u> 0 \quad \text{in } \Omega,\quad u = 0 \quad \mbox{in } \mathbb R^n \setminus\Omega, \tag{ P$_\lambda$} \end{equation*} where $\Omega$ is a bounded domain in $\mathbb{R}^n$ with smooth boundary $\partial \Omega$, $n > 2s$, $ s \in (0,1)$, $ \lambda > 0$, $q> 0$ satisfies $q(2s-1)< (2s+1)$ and $2^*_s=2n/(n-2s)$. Employing the variational method, we show the existence of at least two distinct weak positive solutions for $(\rom{P}_\lambda)$ in $X_0$ when $\lambda \in (0,\Lambda)$ and no solution when $\lambda> \Lambda$, where $\Lambda> 0$ is appropriately chosen. We also prove a result of independent interest that any weak solution to (P$_\lambda)$ is in $C^\alpha(\mathbb R^n)$ with $\alpha=\alpha(s,q)\in (0,1)$. The asymptotic behaviour of weak solutions reveals that this result is sharp.


Fractional Laplacian; very singular nonlinearity; variational method; Hölder regularity

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A. Adimurthi, J. Giacomoni and S. Santra, Positive solutions to a fractional equation with singular nonlinearity, J. Differential Equations 265 (2018), no. 4, 1191–1226.

S. Alama, Semilinear elliptic equations with sublinear indefinite nonlinearities, Adv. Differ. Equ. 4 (1999), 813–842.

D. Applebaum, Lévy process-from probability to finance and quantum groups, Notices Amer. Math. Soc. 51 (2004), 1336–1347.

B. Barrios, E. Colorado, R. Servadei and F. Soria, A critical fractional equation with concave-convex power nonlinearities, Ann. Inst. H. Poincaré 32 (2015), 875–900.

B. Barrios, I. De Bonis, M. Medina and I. Peral, Semilinear problems for the fractional laplacian with a singular nonlinearity, Open Math. 13 (2015), 390–407.

B. Barrios, M. Medina and I. Peral, Some remarks on the solvability of non-local elliptic problems with the Hardy potential, Commun. Contemp. Math. 16 (2014), no. 4, 29 pp., 1350046.

C. Bucur and E. Valdinoci, An Introduction to the Fractional Laplacian, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana, vol. 20, Springer, Cham, 2016.

Z. Cai, C. Chu and C. Lei, Existence of positive solutions for a fractional elliptic problems with the Hardy–Sobolev–Maz’ya potential and critical nonlinearities, Bound. Value Probl. (2017), DOI: 10.1186/s13661-017-0912-8.

W. Chen, Fractional elliptic problems with two critical sobolev-hardy exponents, Electron. J. Differential Equations 22 (2018), 1–12.

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), no. 5, 521–573.

A. Fiscella and P. Pucci, On certain nonlocal Hardy-Sobolev critical elliptic Dirichlet problems, Adv. Differential Equations 21 (2016), no. 5/6, 571–599.

J. Giacomoni, T. Mukherjee and K. Sreenadh, Positive solutions of fractional elliptic equation with critical and singular nonlinearity, Adv. Nonlinear Anal. 6 (2017), no. 6, 327–354.

Y. Haitao, Multiplicity and asymptotic behavior of positive solutions for a singular semilinear elliptic problem, J. Differential Equations 189 (2003), 487–512.

T. Kato, Fractional powers of dissipative operators, J. Math. Soc. Japan 13 (1961), 246–274.

T. Mukherjee and K. Sreenadh, Critical growth fractional elliptic problem with singular nonlinearities, Electron. J. Differential Equations 54 (2016), 1–23.

N. Landkof, Foundations of Modern Potential Theory, Die Grundlehren der Mathematischen Wissenschaften, Band 180, Springer–Verlag, New York, Heidelberg, 1972.

X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl. 101 (2014), 275–302.

R. Servadei and E. Valdinoci, The Brezis–Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc. 367 (2015), 67–102.

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Ph.D. Thesis, The University of Texas at Austin, 2005.

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math. 60 (2007), no. 1, 67–112.

E.M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Math. Ser., vol. 43, Monographs in Harmonic Analysis, III, Princeton University Press, Princeton, NJ, 1993.


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