Rocky Mountain Journal of Mathematics

On nonlocal fractional Laplacian problems with oscillating potentials

Vincenzo Ambrosio, Luigi D'Onofrio, and Giovanni Molica Bisci

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Abstract

In this paper, we deal with the following fractional nonlocal $p$-Laplacian problem: \begin{equation} \begin{cases}(-\Delta )^{s}_{p}u= \lambda \beta (x) u^q + f(u) &\mbox {in } \Omega ,\\ u\geq 0,\ u\not \equiv 0 &\mbox {in } \Omega ,\\ u=0 &\mbox {in } \mathbb{R} ^{N}\setminus \Omega , \end{cases} \end{equation} where $\Omega \subset \mathbb{R} ^{N}$ is a bounded domain with a smooth boun\-dary of $\mathbb{R} ^N$, $s\in (0,1)$, $p\in (1, \infty )$, $N> s p$, $\lambda $ is a real parameter, $\beta \in L^\infty (\Omega )$ is allowed to be indefinite in sign, $q>0$ and $f:[0,+\infty )\to \mathbb{R} $ is a continuous function oscillating near the origin or at infinity. By using variational and topological methods, we obtain the existence of infinitely many solutions for the problem under consideration. The main results obtained here represent some new interesting phenomena in the nonlocal setting.

Article information

Source
Rocky Mountain J. Math., Volume 48, Number 5 (2018), 1399-1436.

Dates
First available in Project Euclid: 19 October 2018

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1539936029

Digital Object Identifier
doi:10.1216/RMJ-2018-48-5-1399

Mathematical Reviews number (MathSciNet)
MR3866552

Zentralblatt MATH identifier
06958785

Subjects
Primary: 35J20: Variational methods for second-order elliptic equations 35J62: Quasilinear elliptic equations 35J92: Quasilinear elliptic equations with p-Laplacian
Secondary: 35J15: Second-order elliptic equations 47J30: Variational methods [See also 58Exx]

Keywords
$p$-fractional Laplacian operator infinitely many solutions variational methods

Citation

Ambrosio, Vincenzo; D'Onofrio, Luigi; Bisci, Giovanni Molica. On nonlocal fractional Laplacian problems with oscillating potentials. Rocky Mountain J. Math. 48 (2018), no. 5, 1399--1436. doi:10.1216/RMJ-2018-48-5-1399. https://projecteuclid.org/euclid.rmjm/1539936029


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References

  • S. Alama and G. Tarantello, Elliptic problems with nonlinearities indefinite in sign, J. Funct. Anal. 141 (1996), 159–215.
    Mathematical Reviews (MathSciNet): MR1414377
    Zentralblatt MATH: 0860.35032
    Digital Object Identifier: doi:10.1006/jfan.1996.0125
  • ––––, On semilinear elliptic equations with indefinite nonlinearities, Calc. Var. Part. Diff. Eqs. 1 (1993), 439–475.
  • B. Barrios, E. Colorado, A. de Pablo and U. Sánchez, On some critical problems for the fractional Laplacian operator, J. Diff. Eqs. 252 (2012), 6133–6162.
  • H. Berestycki, I. Capuzzo Dolcetta and L. Nirenberg, Variational methods for indefinite superlinear homogeneous elliptic problems, Nonlin. Diff. Eqs. Appl. 2 (1995), 553–572.
    Zentralblatt MATH: 0840.35035
    Digital Object Identifier: doi:10.1007/BF01210623
  • H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, 2011.
    Zentralblatt MATH: 1220.46002
  • X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I, Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. Poincare Anal. 31 (2014), 23–53.
  • X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math. 224 (2010), 2052–2093.
  • L. Caffarelli, Non-local diffusions, drifts and games, Nonlin. Part. Diff. Eqs. 7 (2012), 37–52.
    Zentralblatt MATH: 1266.35060
  • L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Part. Diff. Eqs. 32 (2007), 1245–1260.
    Zentralblatt MATH: 1143.26002
    Digital Object Identifier: doi:10.1080/03605300600987306
  • F. Charro, E. Colorado and I. Peral, Multiplicity of solutions to uniformly elliptic fully nonlinear equations with concave-convex right-hand side, J. Diff. Eqs. 246 (2009), 4221–4248.
    Zentralblatt MATH: 1171.35049
    Digital Object Identifier: doi:10.1016/j.jde.2009.01.013
  • D.G. De Figueiredo, J.-P. Gossez and P. Ubilla, Multiplicity results for a family of semilinear elliptic problems under local superlinearity and sublinearity, J. Europ. Math. Soc. 8 (2006), 269–286.
    Zentralblatt MATH: 1245.35048
    Digital Object Identifier: doi:10.4171/JEMS/52
  • ––––, Local superlinearity and sublinearity for indefinite semilinear elliptic problems, J. Funct. Anal. 199 (2003), 452–467.
  • A. Di Castro, T. Kuusi and G. Palatucci, Nonlocal Harnack inequalities, J. Funct. Anal. 267 (2014), 1807–1836.
    Zentralblatt MATH: 1302.35082
    Digital Object Identifier: doi:10.1016/j.jfa.2014.05.023
  • ––––, Local behavior of fractional $p$-minimizers, Ann. Inst. Poincare Anal. 33 (2016), 1279–1299.
  • E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), 521–573.
    Zentralblatt MATH: 1252.46023
    Digital Object Identifier: doi:10.1016/j.bulsci.2011.12.004
  • A. Fiscella, R. Servadei, and E. Valdinoci, Density properties for fractional Sobolev spaces, Ann. Acad. Sci. Fenn. Math. 40 (2015), 235–253.
    Zentralblatt MATH: 1346.46025
    Digital Object Identifier: doi:10.5186/aasfm.2015.4009
  • G. Franzina and G. Palatucci, Fractional $p$-eigenvalues, Riv. Math. Univ. Parma 5 (2014) 373–386.
  • M. Ghergu and V. Rădulescu, Nonlinear PDEs, Mathematical models in biology, chemestry and population genetics, Springer Mono. Math., Springer, Heidelberg, 2012.
  • J. Giacomini, T. Mukherjee and K. Sreenadh, Positive solutions of fractional elliptic equations with critical and singular nonlinearity, Adv. Nonlin. Anal. 6 (2017), 327–354.
  • A. Kristály and Gh. Moroşanu, New competition phenomena in Dirichlet problems, J. Math. Pure. Appl. 94 (2010), 555–570.
  • T. Kuusi, G. Mingione and Y. Sire, Nonlocal equations with measure data, Comm. Math. Phys. 337 (2015), 1317–1368.
    Zentralblatt MATH: 1323.45007
    Digital Object Identifier: doi:10.1007/s00220-015-2356-2
  • E. Lindgren and P. Lindqvist, Fractional eigenvalues, Calc. Var. Part. Diff. Eqs. 49 (2014), 795–826.
    Zentralblatt MATH: 1292.35193
    Digital Object Identifier: doi:10.1007/s00526-013-0600-1
  • J. Mawhin and G. Molica Bisci, A Brezis-Nirenberg type result for a nonlocal fractional operator, J. Lond. Math. Soc. 95 (2017), 79–93.
    Zentralblatt MATH: 06775069
    Digital Object Identifier: doi:10.1112/jlms.12009
  • G. Molica Bisci, Sequence of weak solutions for fractional equations, Math. Res. Lett. 21 (2014), 241–253.
    Zentralblatt MATH: 1298.35239
    Digital Object Identifier: doi:10.4310/MRL.2014.v21.n2.a3
  • G. Molica Bisci and P. Pizzimenti, Sequences of weak solutions for non-local elliptic problems with Dirichlet boundary condition, Proc. Edinburgh Math. Soc. 57 (2014), 779–809.
    Zentralblatt MATH: 1333.35081
    Digital Object Identifier: doi:10.1017/S0013091513000722
  • G. Molica Bisci, V. Rădulescu and R. Servadei, Variational methods for nonlocal fractional problems, Anal. Appl. 15 (2017), 51–82.
  • ––––, Variational methods for nonlocal fractional problems, Encycl. Math. Appl. 162, Cambridge University Press, Cambridge, 2016.
  • G. Molica Bisci and D. Repovš, On doubly nonlocal fractional elliptic equations, Atti Acad. Appl. 26 (2015), 161–176.
  • R. Musina and A. Nazarov, On fractional Laplacians, Comm. Part. Diff. Eqs. 39 (2014), 1780–1790.
    Zentralblatt MATH: 1304.47061
    Digital Object Identifier: doi:10.1080/03605302.2013.864304
  • F. Obersnel and P. Omari, Positive solutions of elliptic problems with locally oscillating nonlinearities, J. Math. Anal. Appl. 323 (2006), 913–929.
    Zentralblatt MATH: 1293.35120
    Digital Object Identifier: doi:10.1016/j.jmaa.2005.11.006
  • P. Omari and F. Zanolin, Infinitely many solutions of a quasilinear elliptic problem with an oscillatory potential, Comm. Part. Diff. Eqs. 21 (1996), 721–733.
    Mathematical Reviews (MathSciNet): MR1391521
    Zentralblatt MATH: 0856.35046
    Digital Object Identifier: doi:10.1080/03605309608821205
  • P. Piersanti and P. Pucci, Existence theorems for fractional $p$-Laplacian problems, Anal. Appl. 15 (2017), 607–640.
    Zentralblatt MATH: 1371.35330
    Digital Object Identifier: doi:10.1142/S0219530516500020
  • P. Pucci and S. Saldi, Critical stationary Kirchhoff equations in $\RR^N$ involving nonlocal operators, Rev. Mat. Iber. 32 (2016), 1–22.
    Zentralblatt MATH: 06569050
    Digital Object Identifier: doi:10.4171/RMI/879
  • ––––, Multiple solutions for an eigenvalue problem involving non-local elliptic $p$-Laplacian operators, in Geometric methods in PDE's, G. Citti, et al., eds., Springer INdAM 11.
  • P. Pucci, M. Xiang and B. Zhang, Multiple solutions for nonhomogeneous Schrodinger-Kirchhoff type equations involving the fractional $p$-Laplacian in $\RR^N$, Calc. Var. Part. Diff. Eqs. 54 (2015), 2785–2806.
  • ––––, Existence and multiplicity of entire solutions for fractional $p$-Kirchhoff equations, Adv. Nonlin. Anal. 5 (2016), 27–55.
  • V. Rădulescu and D. Repovš, Combined effects in nonlinear problems arising in the study of anisotropic continuous media, Nonlin. Anal. 75 (2012), 1524–1530.
  • B. Ricceri, A general variational principle and some of its applications, J. Comp. Appl. Math. 113 (2000), 401–410.
    Zentralblatt MATH: 0946.49001
    Digital Object Identifier: doi:10.1016/S0377-0427(99)00269-1
  • J. Saint Raymond, On the multiplicity of the solutions of the equation $-\Delta u\allowbreak =\allowbreak \lambda f(u)$, J. Diff. Eqs. 180 (2002), 65–88.
  • R. Servadei, Infinitely many solutions for fractional Laplace equations with subcritical nonlinearity, Contemp. Math 595 (2013), 317–340.
    Zentralblatt MATH: 1297.35278
  • R. Servadei and E. Valdinoci, On the spectrum of two different fractional operators, Proc. Roy. Soc. Edinburgh 144 (2014), 1–25.
    Zentralblatt MATH: 1304.35752
    Digital Object Identifier: doi:10.1017/S0308210512001783
  • J. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian, Calc. Var. Part. Diff. Eqs. 36 (2011), 21–41.

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