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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 |
$ (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. $ |
$ \lambda $ |
$ \mu $ |
$ \Omega\subset \mathbb{R}^N $ |
$ s\in (0, 1), N> 2s $ |
$ 0<q<1<r\leq \frac{N+2s}{N- 2s}. $ |
$ (P_ \lambda^\mu) $ |
$ \mu = 0 $ |
$ \lambda_0\in(0, \infty) $ |
$ \lambda> \lambda_0 $ |
$ (P_ \lambda^0) $ |
$ (P_\lambda^0) $ |
$ \lambda>\lambda_0 $ |
$ 0<\mu<\mu_{\lambda} $ |
$ (P_\lambda^\mu) $ |
$ 1<r<\frac{N+2s}{N-2s} $ |
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. |
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. |
[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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