# American Institute of Mathematical Sciences

January  2022, 42(1): 163-187. doi: 10.3934/dcds.2021111

## Combined effects of singular and exponential nonlinearities in fractional Kirchhoff problems

 1 Department of Mathematics, Indian Institute of Technology Jodhpur, Jodhpur, Rajasthan 342037, India 2 Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, 06123 Perugia, Italy 3 College of Science, Civil Aviation University of China, Tianjin, 300300, China

* Corresponding author: Patrizia Pucci

Received  April 2021 Published  January 2022 Early access  August 2021

In this paper we establish the existence of at least two (weak) solutions for the following fractional Kirchhoff problem involving singular and exponential nonlinearities
 $\begin{cases} M\left(\|u\|^{{n}/{s}}\right)(-\Delta)^s_{n/s}u = \mu u^{-q}+ u^{r-1}\exp( u^{\beta})\quad\text{in } \Omega,\\ u>0\qquad\text{in } \Omega,\\ u = 0\qquad\text{in } \mathbb R^n \setminus{ \Omega}, \end{cases}$
where
 $\Omega$
is a smooth bounded domain of
 $\mathbb R^n$
,
 $n\geq 1$
,
 $s\in (0,1)$
,
 $\mu>0$
is a real parameter,
 $\beta <{n/(n-s)}$
and
 $q\in (0,1)$
.The paper covers the so called degenerate Kirchhoff case andthe existence proofs rely on the Nehari manifold techniques.
Citation: Tuhina Mukherjee, Patrizia Pucci, Mingqi Xiang. Combined effects of singular and exponential nonlinearities in fractional Kirchhoff problems. Discrete & Continuous Dynamical Systems, 2022, 42 (1) : 163-187. doi: 10.3934/dcds.2021111
##### References:
 [1] G. Autuori, A. Fiscella and P. Pucci, Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity, Nonlinear Anal., 125 (2015), 699-714.  doi: 10.1016/j.na.2015.06.014.  Google Scholar [2] D. Arcoya and L. M. Mérida, Multiplicity of solutions for a Dirichlet problem with a strongly singular nonlinearity, Nonlinear Anal., 95 (2014), 281-291.  doi: 10.1016/j.na.2013.09.002.  Google Scholar [3] R. Arora, J. Giacomoni, D. Goel and K. Sreenadh, Positive solutions of 1-d half-Laplacian equation with singular and exponential nonlinearity, Asymptot. Anal., 118 (2020), 1-34.  doi: 10.3233/ASY-191557.  Google Scholar [4] K. Bal and P. Garain, Multiplicity of solution for a quasilinear equation with singular nonlinearity, Mediterr. J. Math., 17 (2020) doi: 10.1007/s00009-020-01523-5.  Google Scholar [5] B. Barrios, I. D. Bonis, M. Medina and I. 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Tartar, On a Dirichlet problem with a singular nonlinearity, Comm. Partial Differential Equations, 2 (1977), 193-222.  doi: 10.1080/03605307708820029.  Google Scholar [11] A. Fiscella, A fractional Kirchhoff problem involving a singular term and a critical nonlinearity, Adv. Nonlinear Anal., 8 (2019), 645-660.  doi: 10.1515/anona-2017-0075.  Google Scholar [12] A. Fiscella and P. K. Mishra, The Nehari manifold for fractional Kirchhoff problems involving singular and critical terms, Nonlinear Analysis, 186 (2019), 6-32.  doi: 10.1016/j.na.2018.09.006.  Google Scholar [13] J. Giacomoni, T. Mukherjee and K. Sreenadh, Positive solutions of fractional elliptic equation with critical and singular nonlinearity, Adv. Nonlinear Anal., 6 (2017), 327–354. doi: 10.1515/anona-2016-0113.  Google Scholar [14] J. Giacomoni, T. Mukherjee and K. Sreenadh, A global multiplicity result for a very singular critical nonlocal equation, Topol. Methods Nonlinear Anal., 54 (2019), 345-370.  doi: 10.12775/tmna.2019.049.  Google Scholar [15] J. Giacomoni, I. Schindler and P. Takáč, Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation, Ann. Sc. Norm. Super. Pisa Cl. Sci., 6 (2007), 117–158.  Google Scholar [16] Y. Haitao, Multiplicity and asymptotic behavior of positive solutions for a singular semilinear elliptic problem, J. Differential Equations, 189 (2003), 487-512.  doi: 10.1016/S0022-0396(02)00098-0.  Google Scholar [17] N. Hirano, C. Saccon and N. Shioji, Existence of multiple positive solutions for singular elliptic problems with concave and convex nonlinearities, Adv. Differential Equations, 9 (2004), 197-220.   Google Scholar [18] N. Hirano, C. Saccon and N. Shioji, Brezis-Nirenberg type theorems and multiplicity of positive solutions for a singular elliptic problem, J. Differential Equations, 245 (2008), 1997-2037.  doi: 10.1016/j.jde.2008.06.020.  Google Scholar [19] A. C. Lazer and P. J. McKenna, On a singular nonlinear elliptic boundary-value problem, Proc. Amer. Math. Soc., 111 (1991), 721-730.  doi: 10.1090/S0002-9939-1991-1037213-9.  Google Scholar [20] C.-Y. Lei, J.-F. Liao and C.-L. Tang, Multiple positive solutions for Kirchhoff type of problems with singularity and critical exponents, J. Math. Anal. Appl., 421 (2015), 521-538.  doi: 10.1016/j.jmaa.2014.07.031.  Google Scholar [21] J.-F. Liao, P. Zhang, J. Liu and C.-L. Tang, Existence and multiplicity of positive solutions for a class of Kirchhoff type problems with singularity, J. Math. Anal. Appl., 430 (2015), 1124-1148.  doi: 10.1016/j.jmaa.2015.05.038.  Google Scholar [22] L. Martinazzi, Fractional Adams-Moser-Trudinger type inequalities, Nonlinear Anal., 127 (2015), 263-278.  doi: 10.1016/j.na.2015.06.034.  Google Scholar [23] G. Mingione and V. Rǎdulescu, Recent developments in problems with nonstandard growth and nonuniform ellipticity, J. Math. Anal. Appl., 501 (2021), 125197. doi: 10.1016/j.jmaa.2021.125197.  Google Scholar [24] X. Mingqi, V. D. Rǎdulescu and B. Zhang, Correction to: Fractional Kirchhoff problems with critical Trudinger-Moser nonlinearity, Calc. Var. Partial Differential Equations, 58 (2019), 3pp. doi: 10.1007/s00526-019-1550-z.  Google Scholar [25] X. Mingqi, V. D. Rǎdulescu and B. Zhang, Nonlocal kirchhoff problems with singular exponential nonlinearity, Appl. Math. Optim., (2020). doi: 10.1007/s00245-020-09666-3.  Google Scholar [26] T. Mukherjee and K. Sreenadh, Fractional elliptic equations with critical growth and singular nonlinearities, Electron. J. Differential Equations, 23 (2016), 23pp.  Google Scholar [27] T. Mukherjee and K. Sreenadh, On Dirichlet problem for fractional $p$-Laplacian with singular non-linearity, Adv. Nonlinear Anal., 8 (2019), 52-72.  doi: 10.1515/anona-2016-0100.  Google Scholar [28] L. Wang, K. Cheng and B. Zhang, A uniqueness result for strong singular Kirchhoff-type fractional laplacian problems, Appl. Math. Optim., 83 (2021), 1859-1875.  doi: 10.1007/s00245-019-09612-y.  Google Scholar

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##### References:
 [1] G. Autuori, A. Fiscella and P. Pucci, Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity, Nonlinear Anal., 125 (2015), 699-714.  doi: 10.1016/j.na.2015.06.014.  Google Scholar [2] D. Arcoya and L. M. Mérida, Multiplicity of solutions for a Dirichlet problem with a strongly singular nonlinearity, Nonlinear Anal., 95 (2014), 281-291.  doi: 10.1016/j.na.2013.09.002.  Google Scholar [3] R. Arora, J. Giacomoni, D. Goel and K. Sreenadh, Positive solutions of 1-d half-Laplacian equation with singular and exponential nonlinearity, Asymptot. Anal., 118 (2020), 1-34.  doi: 10.3233/ASY-191557.  Google Scholar [4] K. Bal and P. Garain, Multiplicity of solution for a quasilinear equation with singular nonlinearity, Mediterr. J. Math., 17 (2020) doi: 10.1007/s00009-020-01523-5.  Google Scholar [5] B. Barrios, I. D. Bonis, M. Medina and I. Peral, Semilinear problems for the fractional laplacian with a singular nonlinearity, Open Math., 13 (2015), 390-407.  doi: 10.1515/math-2015-0038.  Google Scholar [6] L. Boccardo, A Dirichlet problem with singular and supercritical nonlinearities, Nonlinear Anal., 75 (2012), 4436-4440.  doi: 10.1016/j.na.2011.09.026.  Google Scholar [7] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011,599pp.  Google Scholar [8] A. Canino, L. Montoro, B. Sciunzi and M. Squassina, Nonlocal problems with singular nonlinearity, Bull. Sci. Math., 141 (2017), 223–250. doi: 10.1016/j.bulsci.2017.01.002.  Google Scholar [9] A. Canino, B. Sciunzi and A. Trombetta, Existence and uniqueness for $p$-Laplace equations involving singular nonlinearities, NoDEA Nonlinear Differential Equations Appl., 23 (2016), 18pp. doi: 10.1007/s00030-016-0361-6.  Google Scholar [10] M. G. Crandall, P. H. Rabinowitz and L. Tartar, On a Dirichlet problem with a singular nonlinearity, Comm. Partial Differential Equations, 2 (1977), 193-222.  doi: 10.1080/03605307708820029.  Google Scholar [11] A. Fiscella, A fractional Kirchhoff problem involving a singular term and a critical nonlinearity, Adv. Nonlinear Anal., 8 (2019), 645-660.  doi: 10.1515/anona-2017-0075.  Google Scholar [12] A. Fiscella and P. K. Mishra, The Nehari manifold for fractional Kirchhoff problems involving singular and critical terms, Nonlinear Analysis, 186 (2019), 6-32.  doi: 10.1016/j.na.2018.09.006.  Google Scholar [13] J. Giacomoni, T. Mukherjee and K. Sreenadh, Positive solutions of fractional elliptic equation with critical and singular nonlinearity, Adv. Nonlinear Anal., 6 (2017), 327–354. doi: 10.1515/anona-2016-0113.  Google Scholar [14] J. Giacomoni, T. Mukherjee and K. Sreenadh, A global multiplicity result for a very singular critical nonlocal equation, Topol. Methods Nonlinear Anal., 54 (2019), 345-370.  doi: 10.12775/tmna.2019.049.  Google Scholar [15] J. Giacomoni, I. Schindler and P. Takáč, Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation, Ann. Sc. Norm. Super. Pisa Cl. Sci., 6 (2007), 117–158.  Google Scholar [16] Y. Haitao, Multiplicity and asymptotic behavior of positive solutions for a singular semilinear elliptic problem, J. Differential Equations, 189 (2003), 487-512.  doi: 10.1016/S0022-0396(02)00098-0.  Google Scholar [17] N. Hirano, C. Saccon and N. Shioji, Existence of multiple positive solutions for singular elliptic problems with concave and convex nonlinearities, Adv. Differential Equations, 9 (2004), 197-220.   Google Scholar [18] N. Hirano, C. Saccon and N. Shioji, Brezis-Nirenberg type theorems and multiplicity of positive solutions for a singular elliptic problem, J. Differential Equations, 245 (2008), 1997-2037.  doi: 10.1016/j.jde.2008.06.020.  Google Scholar [19] A. C. Lazer and P. J. McKenna, On a singular nonlinear elliptic boundary-value problem, Proc. Amer. Math. Soc., 111 (1991), 721-730.  doi: 10.1090/S0002-9939-1991-1037213-9.  Google Scholar [20] C.-Y. Lei, J.-F. Liao and C.-L. Tang, Multiple positive solutions for Kirchhoff type of problems with singularity and critical exponents, J. Math. Anal. Appl., 421 (2015), 521-538.  doi: 10.1016/j.jmaa.2014.07.031.  Google Scholar [21] J.-F. Liao, P. Zhang, J. Liu and C.-L. Tang, Existence and multiplicity of positive solutions for a class of Kirchhoff type problems with singularity, J. Math. Anal. Appl., 430 (2015), 1124-1148.  doi: 10.1016/j.jmaa.2015.05.038.  Google Scholar [22] L. Martinazzi, Fractional Adams-Moser-Trudinger type inequalities, Nonlinear Anal., 127 (2015), 263-278.  doi: 10.1016/j.na.2015.06.034.  Google Scholar [23] G. Mingione and V. Rǎdulescu, Recent developments in problems with nonstandard growth and nonuniform ellipticity, J. Math. Anal. Appl., 501 (2021), 125197. doi: 10.1016/j.jmaa.2021.125197.  Google Scholar [24] X. Mingqi, V. D. Rǎdulescu and B. Zhang, Correction to: Fractional Kirchhoff problems with critical Trudinger-Moser nonlinearity, Calc. Var. Partial Differential Equations, 58 (2019), 3pp. doi: 10.1007/s00526-019-1550-z.  Google Scholar [25] X. Mingqi, V. D. Rǎdulescu and B. Zhang, Nonlocal kirchhoff problems with singular exponential nonlinearity, Appl. Math. Optim., (2020). doi: 10.1007/s00245-020-09666-3.  Google Scholar [26] T. Mukherjee and K. Sreenadh, Fractional elliptic equations with critical growth and singular nonlinearities, Electron. J. Differential Equations, 23 (2016), 23pp.  Google Scholar [27] T. Mukherjee and K. Sreenadh, On Dirichlet problem for fractional $p$-Laplacian with singular non-linearity, Adv. Nonlinear Anal., 8 (2019), 52-72.  doi: 10.1515/anona-2016-0100.  Google Scholar [28] L. Wang, K. Cheng and B. Zhang, A uniqueness result for strong singular Kirchhoff-type fractional laplacian problems, Appl. Math. Optim., 83 (2021), 1859-1875.  doi: 10.1007/s00245-019-09612-y.  Google Scholar
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