Home A critical fractional elliptic equation with singular nonlinearities
Article
Licensed
Unlicensed Requires Authentication

A critical fractional elliptic equation with singular nonlinearities

  • Kamel Saoudi EMAIL logo
Published/Copyright: December 29, 2017
Become an author with De Gruyter Brill
Author Information Explore this Subject
Fractional Calculus and Applied Analysis
From the journal Fractional Calculus and Applied Analysis Volume 20 Issue 6

Abstract

In this work we study the following singular, fractional critical problem

(Pλ)(−Δ)su=λuγ+uqinΩ;u<0inΩ,u=0,inRN∖Ω,

where Ω ⊂ ℝN(N ≥ 3) is a bounded domain with smooth boundary ∂Ω,N < 2s, 0 < 1, λ < 0, 0<γ<1<q≤2s∗−1=N+2sN−2s. Here (–Δ)s is the fractional Laplace operator defined as

(−Δ)su(x)=−12∫RNux+y+ux−y−2u(x)|y|N+2sdy,for allx∈RN.

We use variational methods, in order to show the existence of multiple positive solutions to the problem (Pλ) for different values of λ.

MSC 2010: Primary 35R11; Secondary 35R09; 35A15
Key Words and Phrases: nonlocal operator; fractional Laplacian; singular nonlinearities; variational methods

Appendix A

We start with the following inequality which enables us to estimate when 0 < γ < 1 the singularity in the Gateaux derivative of the energy functional Eλ : X0 → ℝ where the proof can be found in P. Takáč [30].

Lemma A.1

Let 0 < γ < 1. Then, there exists a constant cγ < 0 such that the inequality

∫01∣a+sb∣−γds≤cγ(max0≤γ≤1∣a+sb∣)−γ

holds for all a, b ∈ ℝNwith | a | + | b | > 0.

Now, we study the Gâteaux-differentiability of the energy functional Eλ at a point u ∈ X0.

lemma A.2

Let 0 < γ < 1. Let u ∈ X0such that u ≥ cϕ1. Then ∀ υ ∈ X0, we have

limt→0+1t[Eλ(u+tv)−Eλ(u)]=12∫Q|u(x)−u(y)|(v(x)−v(y))|x−y|N+2sdx dy−∫Ω[λu−γv+uqv]dx.

Proof

Let υ ∈ X0. For sufficiently small t > 0, we have

0≤Eλu+tv−Eλ(u)t=12tu+tv2−u2−λt∫Ω(Gu+tυ−G(u))dx−1t(q+1)∫Ω(|u+tv|q+1−|u|q+1),(A.1)

where

G(u)=11−γ∫Ω(u+)1−γdx  ∀  u∈X0.

We can easily verify that:

(i)(‖u+tυ‖2−‖u‖2)t→2∫Q(u(x)−u(y))(υ(x)−υ(y))|x−y|N+2sdx dy   as    t→0+,(ii)∫Ω(|u+tυ|q+1−|u|q+1)t→(q+1)∫Ω|u|qυ   as    t→0+,

which implies that

(G(u+tυ)−G(u))t∈L1(Ω).

Define, now

g(w)=11−γddw(w+)1−γ{w−γ  if  w>0,0  if  w<0.

Also, for each x ∈ Ω, it follows that

1t(g(u+tv)−g(u))=1t(1−γ)[∫Ω(u+tv)1−γdx−∫Ω(u+)1−γdx]=∫Ω(∫01g(u+stυ)ds) υdx.(A.2)

Then, note that for every x ∈ Ω we have u(x) > 0 and

∫01g(u+stυ)ds→g(u)=u−γ  as  t→0.

Moreover,

|∫01g(u+stυ)ds|≤ ∫01∣u+stυ∣−γds.

Then, using the estimate in Lemma A.1, we get

∫01gu+stυds≤Kγmax0≤s≤1u+stυ−γ≤Kγu−δ≤Kγ(ϵφ1)−γ=Kγ,ϵϕ1−γ,

where the constant Kγ,∈ is a positive constant independent of x ∈ Ω. Moreover, by the Hardy inequality and ∀ υ ∈ X0 we have υϕ1−γ∈L1(Ω). . Finally, using the Lebesgue dominated convergence and letting t → 0+ in (A.2), Lemma A.2 follows. The proof of Lemma A.2 is now completed.  □

Corollary A.1

Let 0 < γ < 1, and1<q≤2s∗−1. Then the energy functional Eλ : X0 → ℝ is Gâteaux-differentiable at every point u ∈ X0that satisfies u ≥ ϕ1in Ω with a constant ε > 0. Its Gâteaux derivativeEλ′(u)at u is given by

Eλ′(u),υ〉=∫Q(u(x)−u(y))(υ(x)−υ(y))|x−y|N+2sdxdy−λ∫Ωu−γυdx−∫Ωuqυdx(A.3)

for υ ∈ X0.

We show now that using a cut-off non-linearity, the associated energy functional is C1 on X0.

Lemma A.3

Let 0 < γ < 1 and υ ∈ X0such that υ ≥ ϵϕ1with some ϵ > 0. Setting for x ∈ Ω,

fλ(x,s)={λs−γ+sq  if  s≥υ(x),λυ(x)−γ+υ(x)q  if  s<υ(x),

Fλ(x,s)=∫0sfλ(x,s)dtand for u ∈ X0,

E¯λ(u)=12∫Ω|u(x)−u(y)|(υ(x)−υ(y))|x−y|N+2s dx dy−∫ΩFλ(x,u)dx

we have that E̅λbelongs toC1(X0, ℝ).

Proof. As in Lemma A.2, we concentrate on the singular term, the others being standard. Let

h(x,s)=s−γifs≥υ(x),υ(x)−γifs<υ(x),

H(x,s)=∫0sh(x,t)dt and S(u) = ∫ΩH(x, u)dx.

First, we determine the Gateaux derivative S′(u). Let υ ∈ X0,

S′(u)υ=limt→0S(u+tυ)−S(u)t=∫Ωmax{u(x),ω(x)}−γυ(x).

Indeed, let U1 = {u(x) + tυ(x) ≥ w(x)}, U2 = {u(x) + tυ(x) < w(x)},V1 = {u(x) ≥ w(x)} and V2 = {u(x) < w(x)}.

Using the above notation, we obtain

S(u+tv)=∫U1[(u+tv(x))1−δ1−δ−ω(x)1−δ1−δ]dx+∫U2ω1−δ(x)dx

and

S(u)=∫V1[u(x)1−δ1−δ−ω(x)1−δ1−δ]dx+∫V2ω1−δ(x)dx.

So, using the previous inequality it follows that

S(u+tυ)−S(u)t=∫U2∩V1((u(x)+tv(x))1−δ1−δ−u(x)1−δ1−δ)dx+∫U1∩V2((u(x)+tυ(x))1−δ−ω(x)1−δ1−δ)dx+∫U2∩V1(ω(x)1−δ1−δ−u(x)1−δ1−δ)dx.

Then, letting t → 0+, we get

limt→0S(u+tυ)−S(u)t=∫V1u−δυ(x)dx+∫V2ω(x)−δυ(x)dx+0.

Now, let uk ∈ X0 such that uk → uλ, it follows that

∣〈S′(uk)−S′(uλ),υ〉∣=|∫Ω(max{uk(x),ω(x)}υ(x)−max{uλ(x),ω(x)}−δυ(x))dx|≤2∫Ωω(x)−δ∣υ(x)∣dx

for all υ ∈ X0. Again, as in Lemma A.2, we use Hardy’s inequality to deduce that φ1−δυ∈L1 , so that by Lesbegue’s dominated convergence theorem we conclude that the Gateaux derivative of S is continuous which implies that S ∈ C1(X0, ℝ). The proof of Lemma A.3 is now completed.  □

Acknowledgements

Thanks are due to the anonymous referees for their careful reading of this paper and useful comments.

References

[1] A. Ambrosetti, H. Brezis, G. Cerami, Combined effects of concave and convexe nonlinearities in some elliptic problems. J. Funct. Anal. 122 (1994), 519–543.10.1006/jfan.1994.1078Search in Google Scholar

[2] A. Ambrosetti, P.H. Rabinowitz, Dual variational methods in critical point theory and applications. J. Funct. Anal. 14 (1973), 349–381.10.1016/0022-1236(73)90051-7Search in Google Scholar

[3] D. Applebaum, Levy Processes and Stochastic Calculus, 2nd ed.. Ser. Cambridge Studies in Adv. Math.116, Cambridge University Press, Cambridge (2009).Search in Google Scholar

[4] D. Averna, S. Tersian, E. Tornatore, on the existence and multiplicity of solutions for Dirichlet’s problem for fractional equations, Fract. Calc. Appl. Anal.19, No 1 (2016), 253–266; 10.1515/fca-2016-0014; https://www.degruyter.com/view7j/fca.2016.19.issue-1/issue-files/fca.2016.19.issue-1.xml.Search in Google Scholar

[5] B. Barrios, I. De Bonis, M. Maria, I. Peral, Semilinear problems for the fractional laplacian with a singular nonlinearity. Open Math. 13 (2015), 390–407.10.1515/math-2015-0038Search in Google Scholar

[6] B. Barriosa, E. Coloradoc, R. Servadeid, F. Soria, A critical fractional equation with concave-convex power nonlinearities. Ann. I. H. Poincare 32 (2015), 875–900.10.1016/j.anihpc.2014.04.003Search in Google Scholar

[7] G. Bonanno, R. Rodrlguez-Lopez, S. Tersian, Existence of solutions to boundary value problem for impulsive fractional differential equations. Fract. Calc. Appl. Anal.17, No 3 (2014), 717–744; 10.2478/s13540-014-0196-y; https://www.degruyter.com/view/j/fca.2014.17.issue-3/issue-files/fca.2014.17.issue-3.xml.Search in Google Scholar

[8] H. Brezis, E. Lieb, A relation between pointwise convergence of functions and convergence of functinals. Proc. Amer. Math. Soc.88, No 3 (1983), 486–490.10.1090/S0002-9939-1983-0699419-3Search in Google Scholar

[9] M.M. Coclite, G. Palmieri, On a singular nonlinear Dirichlet problem. Comm. Partial Differential Equations14, No 10 (1989), 1315–1327.10.1080/03605308908820656Search in Google Scholar

[10] M.G. Crandall, P.H. Rabinowitz, L. Tartar, On a Dirichlet problem with a singular nonlinearity. Comm. Partial Differential Equations2, No 2 (1977), 193–222.10.1080/03605307708820029Search in Google Scholar

[11] A. Cotsiolis, N. Tavoularis, Best constants for Sobolev inequalities for higher order fractional derivatives. J. Math. Anal. Appl. 295 (2004), 225–236.10.1016/j.jmaa.2004.03.034Search in Google Scholar

[12] R. Dhanya, J. Giacomoni, S. Prashanth, K. Saoudi, Global bifurcation and local multiplicity results for elliptic equations with singular nonlinearity of super exponential growth in ℝ2. Advances in Differential Equations17, No 3–4 (2012), 369–400.10.57262/ade/1355703090Search in Google Scholar

[13] Y. Fang, Existence, uniqueness of positive solution to a fractional laplacians with singular non linearity. arXiv Preprint (2014); http://arxiv.org/pdf/1403.3149.pdf.Search in Google Scholar

[14] P. Felmer, A. Quass, Boundary blow up solutions for fractional elliptic equations. Asymptot. Anal.78, No 3 (2012), 123–144.10.3233/ASY-2011-1081Search in Google Scholar

[15] A. Ghanmi, K. Saoudi, The Nehari manifold for a singular elliptic equation involving the fractional Laplace operator. Fractional Differential Calculus6, No 2 (2016), 201–217.10.7153/fdc-06-13Search in Google Scholar

[16] A. Ghanmi, K. Saoudi, A multiplicity results for a singular problem involving the fractional p-Laplacian operator. Complex Var. Elliptic Equ.61, No 9 (2016), 1199–1216.10.1080/17476933.2016.1154548Search in Google Scholar

[17] N. Ghoussoub, D. Preiss, A general mountain pass principle for locating and classifying critical points. Ann. Inst. H. Poincare Anal. Non Lineaire6, No 5 (1989), 321–330.10.1016/S0294-1449(16)30313-4Search in Google Scholar

[18] J. Giacomoni, K. Saoudi, Multiplicity of positive solutions for a singular and critical problem. Nonlinear Anal.71, No 9 (2009), 4060–4077.10.1016/j.na.2009.02.087Search in Google Scholar

[19] Y. Haitao, Multiplicity and asymptotic behavior of positive solutions for a singular semilinear elliptic problem. J. Differential Equations189, No 2 (2003), 487–512.10.1016/S0022-0396(02)00098-0Search in Google Scholar

[20] N. Hirano, C. Saccon, N. Shioji, Existence of multiple positive solutions for singular elliptic problems with concave and convex nonlinearities. Adv. Differential Equations9, No 1–2 (2004), 197–220.10.57262/ade/1355867973Search in Google Scholar

[21] A. Iannizzotto, S. Mosconi, M. Squassina, Hs versus C0-weighted minimizers. Nonlinear Diff. Equations Appl.22, No 3 (2015), 477–497.10.1007/s00030-014-0292-zSearch in Google Scholar

[22] T. Mukherjee, K. Sreenadh, Fractional elliptic equations with critical growth and singular nonlinearities. Electronic J. of Differential Equations 2016, No 54 (2016), 1–23.Search in Google Scholar

[23] R. Rodrlguez-Lopez, S. Tersian, Multiple solutions to boundary value problem for impulsive fractional equations. Fract. Calc. Appl. Anal.17, No 4 (2014), 1016–1038; 10.2478/s13540-014-0212-2; https://www.degruyter.com/viewZj/fca.2014.17.issue-4/issue-files/fca.2014.17.issue-4.xml.Search in Google Scholar

[24] X. Ros-Oton, J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary. J. Math. Pures Appl.101, No 3 (2012), 275–302; 10.1016/j.matpur.2013.06.003.Search in Google Scholar

[25] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator. Commun. Pure Appl. Math.60, No 1 (2007), 67–112.10.1002/cpa.20153Search in Google Scholar

[26] K. Saoudi, M. Kratou, Existence of multiple solutions for a singular and quasilinear equation. Complex Var. Elliptic Equ.60, No 7 (2015), 893–925.10.1080/17476933.2014.981169Search in Google Scholar

[27] R. Servadei, E. Valdinoci, A Brezis-Nirenberg result for nonlocal critical equations in low dimension. Commun. Pure Appl. Anal.12, No 6 (2013), 2445–2464.10.3934/cpaa.2013.12.2445Search in Google Scholar

[28] R. Servadei, E. Valdinoci, Variational methods for non-local operators of elliptic type. Discrete Contin. Dyn. Syst. 33 (2013), 2105–2137.10.3934/dcds.2013.33.2105Search in Google Scholar

[29] R. Servadei, E.Valdinoci, Mountain Pass solutions for non-local elliptic operators. J. Math. Anal. Appl. 389 (2012), 887–898.10.1016/j.jmaa.2011.12.032Search in Google Scholar

[30] P. Takáč, On the Fredholm alternative for the p-Laplacian at the first eigenvalue. Indiana Univ. Math. J.51, No 1 (2002), 187–237.10.1512/iumj.2002.51.2156Search in Google Scholar

[31] S. Yijing, W. Shaoping, L. Yiming, Combined effects of singular and superlinear nonlinearities in some singular boundary value problems. J. Differential Equations176, No 2 (2001), 511–531.10.1006/jdeq.2000.3973Search in Google Scholar

[32] Q.M. Zhou, K.Q. Wang, Existence and multiplicity of solutions for nonlinear elliptic problems with the fractional Laplacian. Fract. Calc. Appl. Anal.18, No 1 (2015), 133–145; 10.1515/fca-2015–0009; https://www.degruyter.com/view/j/fca.2015.18.issue-1/issue-files/fca.2015.18.issue-1.xml.Search in Google Scholar

Received: 2017-3-8
Published Online: 2017-12-29
Published in Print: 2017-12-20

© 2017 Diogenes Co., Sofia

Articles in the same Issue

  1. Frontmatter
  2. Editorial Note
  3. FCAA related news, events and books (FCAA–volume 20–6–2017)
  4. Research Paper
  5. No local L1 solutions for semilinear fractional heat equations
  6. Research Paper
  7. Local and global existence of mild solutions for a class of semilinear fractional integro-differential equations
  8. Research Paper
  9. Wellposedness of Neumann boundary-value problems of space-fractional differential equations
  10. Research Paper
  11. A note on short memory principle of fractional calculus
  12. Research Paper
  13. From fractional order equations to integer order equations
  14. Research Paper
  15. On generalized boundary value problems for a class of fractional differential inclusions
  16. Research Paper
  17. Fractional optimal control problem for variable-order differential systems
  18. Research Paper
  19. Uniqueness of solution for higher-order fractional differential equations with conjugate type integral conditions
  20. Research Paper
  21. Lyapunov-type inequalities for a fractional p-Laplacian system
  22. Research Paper
  23. A critical fractional elliptic equation with singular nonlinearities
  24. Research Paper
  25. A boundary property of some subclasses of functions of bounded type in the half-plane
  26. Research Paper
  27. On weighted generalized fractional and Hardy-type operators acting between Morrey-type spaces
  28. Erratum
  29. Erratum: The mean value theorems and a Nagumo-type uniqueness theorem for Caputo’s fractional calculus
https://doi.org/10.1515/fca-2017-0079
Keywords for this article
nonlocal operator; fractional Laplacian; singular nonlinearities; variational methods

Articles in the same Issue

  1. Frontmatter
  2. Editorial Note
  3. FCAA related news, events and books (FCAA–volume 20–6–2017)
  4. Research Paper
  5. No local L1 solutions for semilinear fractional heat equations
  6. Research Paper
  7. Local and global existence of mild solutions for a class of semilinear fractional integro-differential equations
  8. Research Paper
  9. Wellposedness of Neumann boundary-value problems of space-fractional differential equations
  10. Research Paper
  11. A note on short memory principle of fractional calculus
  12. Research Paper
  13. From fractional order equations to integer order equations
  14. Research Paper
  15. On generalized boundary value problems for a class of fractional differential inclusions
  16. Research Paper
  17. Fractional optimal control problem for variable-order differential systems
  18. Research Paper
  19. Uniqueness of solution for higher-order fractional differential equations with conjugate type integral conditions
  20. Research Paper
  21. Lyapunov-type inequalities for a fractional p-Laplacian system
  22. Research Paper
  23. A critical fractional elliptic equation with singular nonlinearities
  24. Research Paper
  25. A boundary property of some subclasses of functions of bounded type in the half-plane
  26. Research Paper
  27. On weighted generalized fractional and Hardy-type operators acting between Morrey-type spaces
  28. Erratum
  29. Erratum: The mean value theorems and a Nagumo-type uniqueness theorem for Caputo’s fractional calculus
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 27.10.2025 from https://www.degruyterbrill.com/document/doi/10.1515/fca-2017-0079/html
Scroll to top button