# American Institute of Mathematical Sciences

November  2020, 19(11): 5059-5075. doi: 10.3934/cpaa.2020226

## Quasilinear nonlocal elliptic problems with variable singular exponent

 1 Department of Mathematics and Systems Analysis, Aalto University, Espoo - 02150, Finland 2 Department of Mathematics, National Institute of Technology Warangal, Warangal-506004, India

* Corresponding author

Received  February 2020 Revised  April 2020 Published  November 2020 Early access  July 2020

In this article, we provide existence results to the following nonlocal equation
 \begin{align*} \begin{cases} (-\Delta)_p^{s} u = g(x,u),\;u>0\; \text{in}\; \Omega,\\ u = 0 \; \text{in}\; \mathbb{R}^N\setminus \Omega, \end{cases} \end{align*}\quad\quad(P_ \lambda)
where
 $(-\Delta)_{p}^{s}$
is the fractional
 $p$
-Laplacian operator. Here
 $\Omega \subset \mathbb R^N$
is a smooth bounded domain,
 $s\in(0,1)$
,
 $p>1$
and
 $N>sp$
. We establish existence of at least one weak solution for
 $(P_ \lambda)$
when
 $g(x,u) = f(x)u^{-q(x)}$
and existence of at least two weak solutions when
 $g(x,u) = \lambda u^{-q(x)}+ u^{r}$
for a suitable range of
 $\lambda>0$
. Here
 $r\in(p-1,p_{s}^{*}-1)$
where
 $p_s^{*}$
is the critical Sobolev exponent and
 $0 . Citation: Prashanta Garain, Tuhina Mukherjee. Quasilinear nonlocal elliptic problems with variable singular exponent. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5059-5075. doi: 10.3934/cpaa.2020226 ##### References:  [1] V. Ambrosio, Nontrivial solutions for a fractional$p$-Laplacian problem via Rabier theorem, Complex Var. Elliptic Equ., 62 (2017), 838-847. doi: 10.1080/17476933.2016.1245725. [2] V. Ambrosio and T. Isernia, Multiplicity and concentration results for some nonlinear Schrödinger equations with the fractional$p$-Laplacian, Discrete Contin. Dyn. Syst., 38 (2018), 5835-5881. doi: 10.3934/dcds.2018254. [3] D. Arcoya and L. Boccardo, Multiplicity of solutions for a Dirichlet problem with a singular and a supercritical nonlinearities, Differ. Integral Equ., 26 (2013), 119-128. [4] D. Arcoya and L. M. Mérida, Multiplicity of solutions for a Dirichlet problem with a strongly singular nonlinearity, Nonlinear Anal., 95 (2014), 281-291. doi: 10.1016/j.na.2013.09.002. [5] R. Arora, J. Giacomoni, D. Goel and and K. Sreenadh, Positive solutions of 1-D half-laplacian equation with singular and exponential nonlinearity, Asymptotic Anal., 118 (2020), 1-34. doi: 10.3233/ASY-191557. [6] K. Bal and P. Garain, Multiplicity of Solution for a Quasilinear Equation with Singular Nonlinearity, Mediterr. J. Math., 17 (2020), 1-20. doi: 10.1007/s00009-020-01515-5. [7] B. Barrios, I. D. Bonis, M. Medina and I. Peral, Semilinear problems for the fractional laplacian with a singular nonlinearity, Open Math., 13 (2015), 390-407. doi: 10.1515/math-2015-0038. [8] L. Boccardo, A Dirichlet problem with singular and supercritical nonlinearities, Nonlinear Anal., 75 (2012), 4436-4440. doi: 10.1016/j.na.2011.09.026. [9] L. Brasco and E. Parini, The second eigenvalue of the fractional p-laplacian, Adv. Calc. Var., 9 (2016), 323-355. doi: 10.1515/acv-2015-0007. [10] A. Canino, L. Montoro, B. Sciunzi and M. Squassina, Nonlocal problems with singular nonlinearity, Bull. Sci. Math., 141 (2017), 223-250. doi: 10.1016/j.bulsci.2017.01.002. [11] A. Canino, B. Sciunzi and A. Trombetta, Existence and uniqueness for$p$-Laplace equations involving singular nonlinearities, Nonlinear Differ. Equ. Appl., 23 (2016), 1-18. doi: 10.1007/s00030-016-0361-6. [12] J. Carmona and P. J. M. Aparicio, A singular semilinear elliptic equation with a variable exponent, Adv. Nonlinear Stud., 16 (2016), 491-498. doi: 10.1515/ans-2015-5039. [13] M. G. Crandall, P. H. Rabinowitz and L. Tartar, On a Dirichlet problem with a singular nonlinearity, Commun. Partial. Differ. Equ., 2 (1977), 193-222. doi: 10.1080/03605307708820029. [14] L. M. Del Pezzo and A. Quaas, A Hopf's lemma and a strong minimum principle for the fractional$p$-Laplacian, J. Differ. Equ., 263 (2017), 765-778. doi: 10.1016/j.jde.2017.02.051. [15] E. D. Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004. [16] P. Garain and T. Mukherjee, On a class of weighted$p$-Laplace equation with singular nonlinearity, Mediterr. J. Math., 17 (2020), 110. doi: 10.1007/s00009-020-01548-w. [17] P. Garain, On a degenerate singular elliptic problem, Preprint, arXiv: 1803.02102. [18] 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. [19] J. Giacomoni, T. Mukherjee and 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. [20] 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. [21] Y. Haitao, Multiplicity and asymptotic behavior of positive solutions for a singular semilinear elliptic problem, J. Differ. Equ., 189 (2003), 487-512. doi: 10.1016/S0022-0396(02)00098-0. [22] N. Hirano, C. Saccon and N. Shioji, Existence of multiple positive solutions for singular elliptic problems with concave and convex nonlinearities, Adv. Differ. Equ., 9 (2004), 197-220. [23] N. Hirano, C. Saccon and N. Shioji, Brezis-nirenberg type theorems and multiplicity of positive solutions for a singular elliptic problem, J. Differ. Equ., 245 (2008), 1997-2037. doi: 10.1016/j.jde.2008.06.020. [24] A. Iannizzotto, S. Mosconi and M. Squassina, Global hölder regularity for the fractional p-laplacian, Rev. Mat. Iberoam, 32 (2016), 1353-1392. doi: 10.4171/RMI/921. [25] A. C. Lazer and P. J. McKenna, On a singular nonlinear elliptic boundary-value problem, Proc. Amer. Math. Soc., 111 (1991), 721-730. doi: 10.2307/2048410. [26] V. Maźya and T. Shaposhnikova, On the bourgain, brezis, and mironescu theorem concerning limiting embeddings of fractional sobolev spaces, J. Func. Anal., 195 (2002), 230-238. doi: 10.1006/jfan.2002.3955. [27] T. Mukherjee and K. Sreenadh, Fractional elliptic equations with critical growth and singular nonlinearitie, Electron. J. Differ. Equ., 23 (2016), 54. [28] T. Mukherjee and K. Sreenadh, On Dirichlet problem for fractional$p$-Laplacian with singular non-linearity, Adv. Nonlinear Anal., 8 (2019), 52-72. doi: 10.1515/anona-2016-0100. [29] X. R. Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, J. Math. Pure. Appl., 101 (2014), 275-302. doi: 10.1016/j.matpur.2013.06.003. [30] R. Servadei and E. Valdinoci, A Brezis-Nirenberg result for non-local critical equations in low dimension, Commun. Pure Appl. Anal., 12 (2013), 2445-2464. doi: 10.3934/cpaa.2013.12.2445. show all references ##### References:  [1] V. Ambrosio, Nontrivial solutions for a fractional$p$-Laplacian problem via Rabier theorem, Complex Var. Elliptic Equ., 62 (2017), 838-847. doi: 10.1080/17476933.2016.1245725. [2] V. Ambrosio and T. Isernia, Multiplicity and concentration results for some nonlinear Schrödinger equations with the fractional$p$-Laplacian, Discrete Contin. Dyn. Syst., 38 (2018), 5835-5881. doi: 10.3934/dcds.2018254. [3] D. Arcoya and L. Boccardo, Multiplicity of solutions for a Dirichlet problem with a singular and a supercritical nonlinearities, Differ. Integral Equ., 26 (2013), 119-128. [4] D. Arcoya and L. M. Mérida, Multiplicity of solutions for a Dirichlet problem with a strongly singular nonlinearity, Nonlinear Anal., 95 (2014), 281-291. doi: 10.1016/j.na.2013.09.002. [5] R. Arora, J. Giacomoni, D. Goel and and K. Sreenadh, Positive solutions of 1-D half-laplacian equation with singular and exponential nonlinearity, Asymptotic Anal., 118 (2020), 1-34. doi: 10.3233/ASY-191557. [6] K. Bal and P. Garain, Multiplicity of Solution for a Quasilinear Equation with Singular Nonlinearity, Mediterr. J. Math., 17 (2020), 1-20. doi: 10.1007/s00009-020-01515-5. [7] B. Barrios, I. D. Bonis, M. Medina and I. Peral, Semilinear problems for the fractional laplacian with a singular nonlinearity, Open Math., 13 (2015), 390-407. doi: 10.1515/math-2015-0038. [8] L. Boccardo, A Dirichlet problem with singular and supercritical nonlinearities, Nonlinear Anal., 75 (2012), 4436-4440. doi: 10.1016/j.na.2011.09.026. [9] L. Brasco and E. Parini, The second eigenvalue of the fractional p-laplacian, Adv. Calc. Var., 9 (2016), 323-355. doi: 10.1515/acv-2015-0007. [10] A. Canino, L. Montoro, B. Sciunzi and M. Squassina, Nonlocal problems with singular nonlinearity, Bull. Sci. Math., 141 (2017), 223-250. doi: 10.1016/j.bulsci.2017.01.002. [11] A. Canino, B. Sciunzi and A. Trombetta, Existence and uniqueness for$p$-Laplace equations involving singular nonlinearities, Nonlinear Differ. Equ. Appl., 23 (2016), 1-18. doi: 10.1007/s00030-016-0361-6. [12] J. Carmona and P. J. M. Aparicio, A singular semilinear elliptic equation with a variable exponent, Adv. Nonlinear Stud., 16 (2016), 491-498. doi: 10.1515/ans-2015-5039. [13] M. G. Crandall, P. H. Rabinowitz and L. Tartar, On a Dirichlet problem with a singular nonlinearity, Commun. Partial. Differ. Equ., 2 (1977), 193-222. doi: 10.1080/03605307708820029. [14] L. M. Del Pezzo and A. Quaas, A Hopf's lemma and a strong minimum principle for the fractional$p$-Laplacian, J. Differ. Equ., 263 (2017), 765-778. doi: 10.1016/j.jde.2017.02.051. [15] E. D. Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004. [16] P. Garain and T. Mukherjee, On a class of weighted$p$-Laplace equation with singular nonlinearity, Mediterr. J. Math., 17 (2020), 110. doi: 10.1007/s00009-020-01548-w. [17] P. Garain, On a degenerate singular elliptic problem, Preprint, arXiv: 1803.02102. [18] 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. [19] J. Giacomoni, T. Mukherjee and 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. [20] 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. [21] Y. Haitao, Multiplicity and asymptotic behavior of positive solutions for a singular semilinear elliptic problem, J. Differ. Equ., 189 (2003), 487-512. doi: 10.1016/S0022-0396(02)00098-0. [22] N. Hirano, C. Saccon and N. Shioji, Existence of multiple positive solutions for singular elliptic problems with concave and convex nonlinearities, Adv. Differ. Equ., 9 (2004), 197-220. [23] N. Hirano, C. Saccon and N. Shioji, Brezis-nirenberg type theorems and multiplicity of positive solutions for a singular elliptic problem, J. Differ. Equ., 245 (2008), 1997-2037. doi: 10.1016/j.jde.2008.06.020. [24] A. Iannizzotto, S. Mosconi and M. Squassina, Global hölder regularity for the fractional p-laplacian, Rev. Mat. Iberoam, 32 (2016), 1353-1392. doi: 10.4171/RMI/921. [25] A. C. Lazer and P. J. McKenna, On a singular nonlinear elliptic boundary-value problem, Proc. Amer. Math. Soc., 111 (1991), 721-730. doi: 10.2307/2048410. [26] V. Maźya and T. Shaposhnikova, On the bourgain, brezis, and mironescu theorem concerning limiting embeddings of fractional sobolev spaces, J. Func. Anal., 195 (2002), 230-238. doi: 10.1006/jfan.2002.3955. [27] T. Mukherjee and K. Sreenadh, Fractional elliptic equations with critical growth and singular nonlinearitie, Electron. J. Differ. Equ., 23 (2016), 54. [28] 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. [29] X. R. Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, J. Math. Pure. Appl., 101 (2014), 275-302. doi: 10.1016/j.matpur.2013.06.003. [30] R. Servadei and E. Valdinoci, A Brezis-Nirenberg result for non-local critical equations in low dimension, Commun. Pure Appl. Anal., 12 (2013), 2445-2464. doi: 10.3934/cpaa.2013.12.2445.  [1] Yansheng Zhong, Yongqing Li. 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