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  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<q \in C^1(\bar{ \Omega}) $
.
Citation: Prashanta Garain, Tuhina Mukherjee. Quasilinear nonlocal elliptic problems with variable singular exponent. Communications on Pure & Applied Analysis, 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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

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R. AroraJ. GiacomoniD. 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.  Google Scholar

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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.  Google Scholar

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B. BarriosI. D. BonisM. 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

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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

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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.  Google Scholar

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A. CaninoL. MontoroB. 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

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A. CaninoB. 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.  Google Scholar

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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.  Google Scholar

[13]

M. G. CrandallP. 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.  Google Scholar

[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.  Google Scholar

[15]

E. D. NezzaG. 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.  Google Scholar

[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.  Google Scholar

[17]

P. Garain, On a degenerate singular elliptic problem, Preprint, arXiv: 1803.02102. Google Scholar

[18]

J. GiacomoniT. 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

[19]

J. GiacomoniT. 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.  Google Scholar

[20]

J. GiacomoniI. 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

[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.  Google Scholar

[22]

N. HiranoC. 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.   Google Scholar

[23]

N. HiranoC. 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.  Google Scholar

[24]

A. IannizzottoS. Mosconi and M. Squassina, Global hölder regularity for the fractional p-laplacian, Rev. Mat. Iberoam, 32 (2016), 1353-1392.  doi: 10.4171/RMI/921.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

T. Mukherjee and K. Sreenadh, Fractional elliptic equations with critical growth and singular nonlinearitie, Electron. J. Differ. Equ., 23 (2016), 54.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[5]

R. AroraJ. GiacomoniD. 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.  Google Scholar

[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.  Google Scholar

[7]

B. BarriosI. D. BonisM. 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

[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.  Google Scholar

[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.  Google Scholar

[10]

A. CaninoL. MontoroB. 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

[11]

A. CaninoB. 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.  Google Scholar

[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.  Google Scholar

[13]

M. G. CrandallP. 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.  Google Scholar

[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.  Google Scholar

[15]

E. D. NezzaG. 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.  Google Scholar

[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.  Google Scholar

[17]

P. Garain, On a degenerate singular elliptic problem, Preprint, arXiv: 1803.02102. Google Scholar

[18]

J. GiacomoniT. 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

[19]

J. GiacomoniT. 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.  Google Scholar

[20]

J. GiacomoniI. 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

[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.  Google Scholar

[22]

N. HiranoC. 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.   Google Scholar

[23]

N. HiranoC. 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.  Google Scholar

[24]

A. IannizzottoS. Mosconi and M. Squassina, Global hölder regularity for the fractional p-laplacian, Rev. Mat. Iberoam, 32 (2016), 1353-1392.  doi: 10.4171/RMI/921.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

T. Mukherjee and K. Sreenadh, Fractional elliptic equations with critical growth and singular nonlinearitie, Electron. J. Differ. Equ., 23 (2016), 54.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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