# 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 and 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. [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. [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. [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. [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. [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. [7] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011,599pp. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [22] L. Martinazzi, Fractional Adams-Moser-Trudinger type inequalities, Nonlinear Anal., 127 (2015), 263-278.  doi: 10.1016/j.na.2015.06.034. [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. [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. [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. [26] T. Mukherjee and K. Sreenadh, Fractional elliptic equations with critical growth and singular nonlinearities, Electron. J. Differential Equations, 23 (2016), 23pp. [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. [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.

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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. [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. [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. [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. [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. [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. [7] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011,599pp. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [22] L. Martinazzi, Fractional Adams-Moser-Trudinger type inequalities, Nonlinear Anal., 127 (2015), 263-278.  doi: 10.1016/j.na.2015.06.034. [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. [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. [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. [26] T. Mukherjee and K. Sreenadh, Fractional elliptic equations with critical growth and singular nonlinearities, Electron. J. Differential Equations, 23 (2016), 23pp. [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. [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.
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