# American Institute of Mathematical Sciences

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

## A multiparameter fractional Laplace problem with semipositone nonlinearity

 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 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
, we show the existence of second positive solution via mountain pass argument.
Citation: R. Dhanya, Sweta Tiwari. A multiparameter fractional Laplace problem with semipositone nonlinearity. Communications on Pure and 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [11] R. Dhanya, Positive solution curves of an infinite semipositone problem, Electron. J. Differ. Equ., 2018 (2018), 1-14. [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. [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. [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. [15] G. Franzina and G. Palatucci, Fractional $p$-eigenvalues, Riv. Math. Univ. Parma (N.S.), 5 (2014), 373–386. [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. [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. [18] P. L. Lions, On the existence of positive solutions of semilinear elliptic equations, Siam Review, 24 (1982), 441-467.  doi: 10.1137/1024101. [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. [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. [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. [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. [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. [24] Xavier Ros-Oton, Nonlocal elliptic equations in bounded domains: a survey, Publ. Mat., 60 (2016), 3-26. [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. [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.

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##### 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [11] R. Dhanya, Positive solution curves of an infinite semipositone problem, Electron. J. Differ. Equ., 2018 (2018), 1-14. [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. [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. [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. [15] G. Franzina and G. Palatucci, Fractional $p$-eigenvalues, Riv. Math. Univ. Parma (N.S.), 5 (2014), 373–386. [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. [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. [18] P. L. Lions, On the existence of positive solutions of semilinear elliptic equations, Siam Review, 24 (1982), 441-467.  doi: 10.1137/1024101. [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. [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. [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. [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. [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. [24] Xavier Ros-Oton, Nonlocal elliptic equations in bounded domains: a survey, Publ. Mat., 60 (2016), 3-26. [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. [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.
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