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doi: 10.3934/cpaa.2021143
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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 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 &amp; Applied Analysis, 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

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