American Institute of Mathematical Sciences

Advanced
December  2021, 20(12): 4043-4061. doi: 10.3934/cpaa.2021143

A multiparameter fractional Laplace problem with semipositone nonlinearity

R. Dhanya 1, and  Sweta Tiwari 2,,

1. 

Indian Institute of Science Education and Research, Thiruvananthapuram, Thiruvananthapuram 695551, India
2. 

Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati-781039, India

* Corresponding author

Received  February 2021 Revised  June 2021 Published  December 2021 Early access  August 2021

Fund Project: R. Dhanya was supported by INSPIRE faculty fellowship (DST/INSPIRE/04/2015/003221) when the work was being carried out

In this paper we prove the existence of at least one positive solution for nonlocal semipositone problem of the type
$ (P_\lambda^\mu)\left\{ \begin{array}{rcl} (-\Delta)^s u& = & \lambda(u^{q}-1)+\mu u^r \text{ in } \Omega\\ u&>&0 \text{ in } \Omega\\ u&\equiv &0 \text{ on }{\mathbb R^N\setminus\Omega}. \end{array}\right. $
when the positive parameters
$ \lambda $
 and
$ \mu $
 belong to certain range. Here
$ \Omega\subset \mathbb{R}^N $
 is assumed to be a bounded open set with smooth boundary,
$ s\in (0, 1), N> 2s $
 and
$ 0<q<1<r\leq \frac{N+2s}{N- 2s}. $
 First we consider
$ (P_ \lambda^\mu) $
 when
$ \mu = 0 $
 and prove that there exists
$ \lambda_0\in(0, \infty) $
 such that for all
$ \lambda> \lambda_0 $
 the problem
$ (P_ \lambda^0) $
 admits at least one positive solution. In fact we will show the existence of a continuous branch of maximal solutions of
$ (P_\lambda^0) $
 emanating from infinity. Next for each
$ \lambda>\lambda_0 $
 and for all
$ 0<\mu<\mu_{\lambda} $
 we establish the existence of at least one positive solution of
$ (P_\lambda^\mu) $
 using variational method. Also in the sub critical case, i.e., for
$ 1<r<\frac{N+2s}{N-2s} $
, we show the existence of second positive solution via mountain pass argument.
Keywords: Non-local problem, fractional laplacian, semipositone problem, indefinite nonlinearities, critical exponent.
Mathematics Subject Classification: Primary: 35A15, 35B33, 35R11; Secondary: 35J20.
Citation: R. Dhanya, Sweta Tiwari. A multiparameter fractional Laplace problem with semipositone nonlinearity. Communications on Pure & Applied Analysis, 2021, 20 (12) : 4043-4061. doi: 10.3934/cpaa.2021143
References:
[1]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[2]

H. Berestycki, L. A. Caffarelli and L. Nirenberg, Monotonicity for elliptic equations in unbounded Lipschitz domains, Comm. Pure Appl. Math., 50 (1997), 1089-1111.  doi: 10.1002/(SICI)1097-0312(199711)50:11<1089::AID-CPA2>3.0.CO;2-6.  Google Scholar

[3] G. M. Bisci, V. D. Rǎdulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems: Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2016.  doi: 10.1017/CBO9781316282397.  Google Scholar
[4]

G. M. Bisci and R. Servadei, A bifurcation result for non-local fractional equations, Anal. Appl. (Singap.), 13 (2015), 371-394.  doi: 10.1142/S0219530514500067.  Google Scholar

[5]

G. M. Bisci and R. Servadei, Lower semicontinuity of functionals of fractional type and applications to nonlocal equations with critical Sobolev exponent, Adv. Differ. Equ., 20 (2015), 635-660.   Google Scholar

[6]

H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.  doi: 10.2307/2044999.  Google Scholar

[7]

C. Bucur and E. Valdinoci, Nonlocal diffusion and applications, Lecture Notes of the Unione Matematica Italiana, 20. Springer. doi: 10.1007/978-3-319-28739-3.  Google Scholar

[8]

A. Castro and R. Shivaji, Nonnegative solutions for a class of nonpositone problems, Proc. Roy. Soc. Edinburgh Sect. A, 108 (1988), 291-302.  doi: 10.1017/S0308210500014670.  Google Scholar

[9]

David G. Costa, Humberto Ramos Quoirin and Jianfu Yang, On a variational approach to existence and multiplicity results for semipositone problems, Electronic J. Differ. Equ., 2006 (2006), 1-10.   Google Scholar

[10]

David G. Costa, Humberto Ramos Quoirin and Hossein Tehrani, A Variational approach to superliner semipositone elliptic problems, Proc. Amer. Math. Soc., 145 (2017), 2661-2675.  doi: 10.1090/proc/13426.  Google Scholar

[11]

R. Dhanya, Positive solution curves of an infinite semipositone problem, Electron. J. Differ. Equ., 2018 (2018), 1-14.   Google Scholar

[12]

R. Dhanya, Q. Morris and R. Shivaji, Existence of positive radial solutions for superlinear, semipositone problems on the exterior of a ball, J Math. Anal. Appl., 434 (2016), 1533-1548.  doi: 10.1016/j.jmaa.2015.07.016.  Google Scholar

[13]

A. Fiscella, R. Servadei and E. Valdinoci, Density properties for fractional Sobolev spaces, Ann. Acad. Sci. Fenn. Math., 40 (2015), 235-253.  doi: 10.5186/aasfm.2015.4009.  Google Scholar

[14]

Francesca Faraci and Csaba Farkas, A quasilinear elliptic problem involving critical Sobolev exponent, Collect. Math., 66 (2015), 243-259.  doi: 10.1007/s13348-014-0125-8.  Google Scholar

[15]

G. Franzina and G. Palatucci, Fractional $p$-eigenvalues, Riv. Math. Univ. Parma (N.S.), 5 (2014), 373–386.  Google Scholar

[16]

Jacques J. Giacomoni, Tuhina Mukherjee and Konijeti Sreenadh, Existence of three positive solutions for a nonlocal singular Dirichlet boundary problem, Adv. Nonlinear Stud., 19 (2019), 333-352.  doi: 10.1515/ans-2018-0011.  Google Scholar

[17]

Tommaso Leonori, Ireneo Peral, Ana Primo and Fernando Soria, Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations, Discrete Contin. Dyn. Syst., 35 (2015), 6031-6068.  doi: 10.3934/dcds.2015.35.6031.  Google Scholar

[18]

P. L. Lions, On the existence of positive solutions of semilinear elliptic equations, Siam Review, 24 (1982), 441-467.  doi: 10.1137/1024101.  Google Scholar

[19]

J. Mawhin and M. Bisci, A Brezis-Nirenberg type result for a nonlocal fractional operator, J. London Math. Soc., 95, (2017), 73–93. doi: 10.1112/jlms.12009.  Google Scholar

[20]

Quinn Morris, Ratnasingham Shivaji and Inbo Sim, Existence of positive radial solutions for a superlinear semipo sitone p-Laplacian problem on the exterior of a ball, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 148 (2018), 409-428.  doi: 10.1017/S0308210517000452.  Google Scholar

[21]

Eleonora Di Nezza, Giampiero Palatucci and Enrico Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bulletin des Sciences Mathématiques, 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[22]

K. Perera, R. Shivaji and I. Sim, A class of semipositone p-Laplacian problems with a critical growth reaction term, Adv. Nonlinear Anal., 9 (2020), 516-525.  doi: 10.1515/anona-2020-0012.  Google Scholar

[23]

K. Perera and R. Shivaji, Positive solutions of multiparameter semipositone $p$-Laplacian problems, J. Math. Anal. Appl., 338 (2008), 1397-1400.  doi: 10.1016/j.jmaa.2007.05.085.  Google Scholar

[24]

Xavier Ros-Oton, Nonlocal elliptic equations in bounded domains: a survey, Publ. Mat., 60 (2016), 3-26.   Google Scholar

[25]

Xavier Ros-Oton and Joaquim Serra, The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302.  doi: 10.1016/j.matpur.2013.06.003.  Google Scholar

[26]

M. Squassina, Two solutions for inhomogeneous nonlinear elliptic equations at critical growth, Nonlinear Differ. Equ. Appl., 11 (2004), 53-71.  doi: 10.1007/s00030-003-1046-5.  Google Scholar

show all references

References:
[1]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[2]

H. Berestycki, L. A. Caffarelli and L. Nirenberg, Monotonicity for elliptic equations in unbounded Lipschitz domains, Comm. Pure Appl. Math., 50 (1997), 1089-1111.  doi: 10.1002/(SICI)1097-0312(199711)50:11<1089::AID-CPA2>3.0.CO;2-6.  Google Scholar

[3] G. M. Bisci, V. D. Rǎdulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems: Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2016.  doi: 10.1017/CBO9781316282397.  Google Scholar
[4]

G. M. Bisci and R. Servadei, A bifurcation result for non-local fractional equations, Anal. Appl. (Singap.), 13 (2015), 371-394.  doi: 10.1142/S0219530514500067.  Google Scholar

[5]

G. M. Bisci and R. Servadei, Lower semicontinuity of functionals of fractional type and applications to nonlocal equations with critical Sobolev exponent, Adv. Differ. Equ., 20 (2015), 635-660.   Google Scholar

[6]

H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.  doi: 10.2307/2044999.  Google Scholar

[7]

C. Bucur and E. Valdinoci, Nonlocal diffusion and applications, Lecture Notes of the Unione Matematica Italiana, 20. Springer. doi: 10.1007/978-3-319-28739-3.  Google Scholar

[8]

A. Castro and R. Shivaji, Nonnegative solutions for a class of nonpositone problems, Proc. Roy. Soc. Edinburgh Sect. A, 108 (1988), 291-302.  doi: 10.1017/S0308210500014670.  Google Scholar

[9]

David G. Costa, Humberto Ramos Quoirin and Jianfu Yang, On a variational approach to existence and multiplicity results for semipositone problems, Electronic J. Differ. Equ., 2006 (2006), 1-10.   Google Scholar

[10]

David G. Costa, Humberto Ramos Quoirin and Hossein Tehrani, A Variational approach to superliner semipositone elliptic problems, Proc. Amer. Math. Soc., 145 (2017), 2661-2675.  doi: 10.1090/proc/13426.  Google Scholar

[11]

R. Dhanya, Positive solution curves of an infinite semipositone problem, Electron. J. Differ. Equ., 2018 (2018), 1-14.   Google Scholar

[12]

R. Dhanya, Q. Morris and R. Shivaji, Existence of positive radial solutions for superlinear, semipositone problems on the exterior of a ball, J Math. Anal. Appl., 434 (2016), 1533-1548.  doi: 10.1016/j.jmaa.2015.07.016.  Google Scholar

[13]

A. Fiscella, R. Servadei and E. Valdinoci, Density properties for fractional Sobolev spaces, Ann. Acad. Sci. Fenn. Math., 40 (2015), 235-253.  doi: 10.5186/aasfm.2015.4009.  Google Scholar

[14]

Francesca Faraci and Csaba Farkas, A quasilinear elliptic problem involving critical Sobolev exponent, Collect. Math., 66 (2015), 243-259.  doi: 10.1007/s13348-014-0125-8.  Google Scholar

[15]

G. Franzina and G. Palatucci, Fractional $p$-eigenvalues, Riv. Math. Univ. Parma (N.S.), 5 (2014), 373–386.  Google Scholar

[16]

Jacques J. Giacomoni, Tuhina Mukherjee and Konijeti Sreenadh, Existence of three positive solutions for a nonlocal singular Dirichlet boundary problem, Adv. Nonlinear Stud., 19 (2019), 333-352.  doi: 10.1515/ans-2018-0011.  Google Scholar

[17]

Tommaso Leonori, Ireneo Peral, Ana Primo and Fernando Soria, Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations, Discrete Contin. Dyn. Syst., 35 (2015), 6031-6068.  doi: 10.3934/dcds.2015.35.6031.  Google Scholar

[18]

P. L. Lions, On the existence of positive solutions of semilinear elliptic equations, Siam Review, 24 (1982), 441-467.  doi: 10.1137/1024101.  Google Scholar

[19]

J. Mawhin and M. Bisci, A Brezis-Nirenberg type result for a nonlocal fractional operator, J. London Math. Soc., 95, (2017), 73–93. doi: 10.1112/jlms.12009.  Google Scholar

[20]

Quinn Morris, Ratnasingham Shivaji and Inbo Sim, Existence of positive radial solutions for a superlinear semipo sitone p-Laplacian problem on the exterior of a ball, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 148 (2018), 409-428.  doi: 10.1017/S0308210517000452.  Google Scholar

[21]

Eleonora Di Nezza, Giampiero Palatucci and Enrico Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bulletin des Sciences Mathématiques, 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[22]

K. Perera, R. Shivaji and I. Sim, A class of semipositone p-Laplacian problems with a critical growth reaction term, Adv. Nonlinear Anal., 9 (2020), 516-525.  doi: 10.1515/anona-2020-0012.  Google Scholar

[23]

K. Perera and R. Shivaji, Positive solutions of multiparameter semipositone $p$-Laplacian problems, J. Math. Anal. Appl., 338 (2008), 1397-1400.  doi: 10.1016/j.jmaa.2007.05.085.  Google Scholar

[24]

Xavier Ros-Oton, Nonlocal elliptic equations in bounded domains: a survey, Publ. Mat., 60 (2016), 3-26.   Google Scholar

[25]

Xavier Ros-Oton and Joaquim Serra, The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302.  doi: 10.1016/j.matpur.2013.06.003.  Google Scholar

[26]

M. Squassina, Two solutions for inhomogeneous nonlinear elliptic equations at critical growth, Nonlinear Differ. Equ. Appl., 11 (2004), 53-71.  doi: 10.1007/s00030-003-1046-5.  Google Scholar

[1]

Xudong Shang, Jihui Zhang, Yang Yang. Positive solutions of nonhomogeneous fractional Laplacian problem with critical exponent. Communications on Pure & Applied Analysis, 2014, 13 (2) : 567-584. doi: 10.3934/cpaa.2014.13.567
[2]

Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253
[3]

Antonio Capella. Solutions of a pure critical exponent problem involving the half-laplacian in annular-shaped domains. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1645-1662. doi: 10.3934/cpaa.2011.10.1645
[4]

Wenxiong Chen, Congming Li, Jiuyi Zhu. Fractional equations with indefinite nonlinearities. Discrete & Continuous Dynamical Systems, 2019, 39 (3) : 1257-1268. doi: 10.3934/dcds.2019054
[5]

Yanan Li, Alexandre N. Carvalho, Tito L. M. Luna, Estefani M. Moreira. A non-autonomous bifurcation problem for a non-local scalar one-dimensional parabolic equation. Communications on Pure & Applied Analysis, 2020, 19 (11) : 5181-5196. doi: 10.3934/cpaa.2020232
[6]

Yansheng Zhong, Yongqing Li. On a p-Laplacian eigenvalue problem with supercritical exponent. Communications on Pure & Applied Analysis, 2019, 18 (1) : 227-236. doi: 10.3934/cpaa.2019012
[7]

Joseph G. Conlon, André Schlichting. A non-local problem for the Fokker-Planck equation related to the Becker-Döring model. Discrete & Continuous Dynamical Systems, 2019, 39 (4) : 1821-1889. doi: 10.3934/dcds.2019079
[8]

Christos V. Nikolopoulos, Georgios E. Zouraris. Numerical solution of a non-local elliptic problem modeling a thermistor with a finite element and a finite volume method. Conference Publications, 2007, 2007 (Special) : 768-778. doi: 10.3934/proc.2007.2007.768
[9]

Monica Marras, Nicola Pintus, Giuseppe Viglialoro. On the lifespan of classical solutions to a non-local porous medium problem with nonlinear boundary conditions. Discrete & Continuous Dynamical Systems - S, 2020, 13 (7) : 2033-2045. doi: 10.3934/dcdss.2020156
[10]

Qingfang Wang. The Nehari manifold for a fractional Laplacian equation involving critical nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2261-2281. doi: 10.3934/cpaa.2018108
[11]

Mikko Kemppainen, Peter Sjögren, José Luis Torrea. Wave extension problem for the fractional Laplacian. Discrete & Continuous Dynamical Systems, 2015, 35 (10) : 4905-4929. doi: 10.3934/dcds.2015.35.4905
[12]

Maoding Zhen, Jinchun He, Haoyuan Xu, Meihua Yang. Positive ground state solutions for fractional Laplacian system with one critical exponent and one subcritical exponent. Discrete & Continuous Dynamical Systems, 2019, 39 (11) : 6523-6539. doi: 10.3934/dcds.2019283
[13]

Jinguo Zhang, Dengyun Yang. Fractional $ p $-sub-Laplacian operator problem with concave-convex nonlinearities on homogeneous groups. Electronic Research Archive, 2021, 29 (5) : 3243-3260. doi: 10.3934/era.2021036
[14]

Thierry Horsin, Mohamed Ali Jendoubi. Asymptotics for some discretizations of dynamical systems, application to second order systems with non-local nonlinearities. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2022007
[15]

Qi-Lin Xie, Xing-Ping Wu, Chun-Lei Tang. Existence and multiplicity of solutions for Kirchhoff type problem with critical exponent. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2773-2786. doi: 10.3934/cpaa.2013.12.2773
[16]

Imran H. Biswas, Indranil Chowdhury. On the differentiability of the solutions of non-local Isaacs equations involving $\frac{1}{2}$-Laplacian. Communications on Pure & Applied Analysis, 2016, 15 (3) : 907-927. doi: 10.3934/cpaa.2016.15.907
[17]

Li Li. An inverse problem for a fractional diffusion equation with fractional power type nonlinearities. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021064
[18]

Rui-Qi Liu, Chun-Lei Tang, Jia-Feng Liao, Xing-Ping Wu. Positive solutions of Kirchhoff type problem with singular and critical nonlinearities in dimension four. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1841-1856. doi: 10.3934/cpaa.2016006
[19]

Raffaella Servadei, Enrico Valdinoci. A Brezis-Nirenberg result for non-local critical equations in low dimension. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2445-2464. doi: 10.3934/cpaa.2013.12.2445
[20]

Keyan Wang. Global well-posedness for a transport equation with non-local velocity and critical diffusion. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1203-1210. doi: 10.3934/cpaa.2008.7.1203

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (154)
  • HTML views (215)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy