January  2017, 16(1): 243-252. doi: 10.3934/cpaa.2017011

Trudinger-Moser type inequality and existence of solution for perturbed non-local elliptic operators with exponential nonlinearity

Mathematics Department, University of Monastir, Faculty of Sciences, Monastir, 5019, Tunisia

Received  April 2016 Revised  August 2016 Published  November 2016

In this paper we consider the following perturbed nonlocal problem with exponential nonlinearity
$\begin{cases}-\mathcal{L}_{K}u+ \left|u\right|^{p-2}u+h(u)= f \ \ \ \ \ \mbox{in} \ \ \Omega,\\u=0, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{in}\ \ \mathbb{R}^{N}\setminus \Omega,\end{cases}$
where $s\in (0, 1)$, $N=ps$, $p\geq 2$ and $f\in L.{\infty}(\mathbb{R}^{N})$. First, we generalize a suitable Trudinger-Moser inequality to a fractional functional space. Then, using the Ekeland's variational principle, we prove the existence of a solution of problem (1).
Citation: Anouar Bahrouni. Trudinger-Moser type inequality and existence of solution for perturbed non-local elliptic operators with exponential nonlinearity. Communications on Pure and Applied Analysis, 2017, 16 (1) : 243-252. doi: 10.3934/cpaa.2017011
References:
[1]

S. Adachi and K. Tanaka, Trudinger type inequalities in ${\mathbb{R}.n}$ and their best constants, Proc. Amer. Math. Soc, 128 (2000), 2051-2057. doi: 10.1090/S0002-9939-99-05180-1.

[2]

G. Autuori and P. Pucci, Elliptic problems involving the fractional Laplacian in ${\mathbb{R}.n}$, J. Differential Equations, 255 (2012), 2340-2362. doi: 10.1016/j.jde.2013.06.016.

[3]

Giovanni M. Bisci and V. Radulescu, Ground state solutions of scalar field fractional Schrödinger equations, Calc. Var., 54 (2015), 2985-3008. doi: 10.1007/s00526-015-0891-5.

[4]

G. Molica Bisci, V. Radulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Combridge University press, Cambridge, 2015. doi: 10.1017/CBO9781316282397.

[5]

Haim Brezis and Frank Merle, Uniform estimates and blow-up behavior for solutions of -∆u = V(x)eu in two dimensions, Comm. Partial Differential Equations, 16 (1991), 1223-1253. doi: 10.1080/03605309108820797.

[6]

L. Cafarelli, S. Salsa and L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplatian, Invent. Math, 171 (2008), 425-461. doi: 10.1007/s00222-007-0086-6.

[7]

L. Caffarelli, J-M Roque joffre and Y. Sire, Variational problems for free boundaries for the fractional Laplacian, J. Eur. Math. Soc, 12 (2010), 1151-1179. doi: 10.4171/JEMS/226.

[8]

X. Cheng, Ground state solutions of asymptotically linear fractional Schrödinger equations with unbounded potential, J. Math. Phys, 54 (2013), 061504. doi: 10.1063/1.4809933.

[9]

I. Ekeland, On the variational principle principle, J. Math. Anal. Appl, 47 (2014), 324-353.

[10]

P. Felmer, A. Quaas and J. Tan, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Pro. Roy. Soc. Edinburgh Sect A, 142 (2012), 1237-1262.

[11]

D. G. de Figueiredo, J. M. do Ò, B. Ruf, On an inequality by N. Trudinger and J. Moser and related elliptic equations, Commun. Pure Appl. Math, 55 (2002), 135-152. doi: 10.1002/cpa.10015.

[12]

D. G. de Figueiredo, O. H. Miyagaki and B. Ruf, Elliptic equations in R2 with nonlinearities in he critical growth range, Calc. Var. Partial Differ. Equ, 3 (1995), 139-153. doi: 10.1007/BF01205003.

[13]

A. Fiscella, R. Servadei and E. Valdinoci, Density properties for fractional Sobolev spaces, preprint. doi: 10.5186/aasfm.2015.4009.

[14]

G. Franzina and G. Palatucci, Fractional p-eigenvalues, Riv. Mat. Univ. Parma, 5 (2014), 373-386.

[15]

A. Iannizzotto and M. Squassina, $\frac{1}{2}$ -Laplacian problems with exponential nonlinearity, J. Math. Anal. Appl, 414 (2014), 372-385. doi: 10.1016/j.jmaa.2013.12.059.

[16]

Yan Yan Li and Itai Shafrir, Blow-up analysis for solutions of -∆u = Veu in dimension two, Indiana Univ. Math. J, 43 (1994), 1255-1270. doi: 10.1512/iumj.1994.43.43054.

[17]

E. Di 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.

[18]

T. Ozawa, On critical cases of Sobolev's inequalities, J. Funct. Analysis, 127 (1995), 259-269. doi: 10.1006/jfan.1995.1012.

[19]

R. Servadei and E. Valdinoci, Mountain Pass solutions for non-local elliptic operators, J. Math. Anal. Appl, 389 (2012), 887-898. doi: 10.1016/j.jmaa.2011.12.032.

[20]

R. Servadei and E. Valdinoci, Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators, Rev. Mat. Iberoam, 29 (2013), 1091-1126. doi: 10.4171/RMI/750.

[21]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst, 33 (2013), 2105-2137.

[22]

Y. Sire and E. Valdinoci, Rigidity results for some boundary quasilinear phase transitions, Comm. Partial. Differ. Eq, 33 (2009), 765-784. doi: 10.1080/03605300902892402.

[23]

N. S. Trudinger, On imbeddings into Orcliz spaces and some applications, J. Math. Mech, 17 (1967), 473-483.

[24]

J. L. Vázquez, Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators, Discrete Contin. Dyn. Syst, 7 (2014), 857-885. doi: 10.3934/dcdss.2014.7.857.

show all references

References:
[1]

S. Adachi and K. Tanaka, Trudinger type inequalities in ${\mathbb{R}.n}$ and their best constants, Proc. Amer. Math. Soc, 128 (2000), 2051-2057. doi: 10.1090/S0002-9939-99-05180-1.

[2]

G. Autuori and P. Pucci, Elliptic problems involving the fractional Laplacian in ${\mathbb{R}.n}$, J. Differential Equations, 255 (2012), 2340-2362. doi: 10.1016/j.jde.2013.06.016.

[3]

Giovanni M. Bisci and V. Radulescu, Ground state solutions of scalar field fractional Schrödinger equations, Calc. Var., 54 (2015), 2985-3008. doi: 10.1007/s00526-015-0891-5.

[4]

G. Molica Bisci, V. Radulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Combridge University press, Cambridge, 2015. doi: 10.1017/CBO9781316282397.

[5]

Haim Brezis and Frank Merle, Uniform estimates and blow-up behavior for solutions of -∆u = V(x)eu in two dimensions, Comm. Partial Differential Equations, 16 (1991), 1223-1253. doi: 10.1080/03605309108820797.

[6]

L. Cafarelli, S. Salsa and L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplatian, Invent. Math, 171 (2008), 425-461. doi: 10.1007/s00222-007-0086-6.

[7]

L. Caffarelli, J-M Roque joffre and Y. Sire, Variational problems for free boundaries for the fractional Laplacian, J. Eur. Math. Soc, 12 (2010), 1151-1179. doi: 10.4171/JEMS/226.

[8]

X. Cheng, Ground state solutions of asymptotically linear fractional Schrödinger equations with unbounded potential, J. Math. Phys, 54 (2013), 061504. doi: 10.1063/1.4809933.

[9]

I. Ekeland, On the variational principle principle, J. Math. Anal. Appl, 47 (2014), 324-353.

[10]

P. Felmer, A. Quaas and J. Tan, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Pro. Roy. Soc. Edinburgh Sect A, 142 (2012), 1237-1262.

[11]

D. G. de Figueiredo, J. M. do Ò, B. Ruf, On an inequality by N. Trudinger and J. Moser and related elliptic equations, Commun. Pure Appl. Math, 55 (2002), 135-152. doi: 10.1002/cpa.10015.

[12]

D. G. de Figueiredo, O. H. Miyagaki and B. Ruf, Elliptic equations in R2 with nonlinearities in he critical growth range, Calc. Var. Partial Differ. Equ, 3 (1995), 139-153. doi: 10.1007/BF01205003.

[13]

A. Fiscella, R. Servadei and E. Valdinoci, Density properties for fractional Sobolev spaces, preprint. doi: 10.5186/aasfm.2015.4009.

[14]

G. Franzina and G. Palatucci, Fractional p-eigenvalues, Riv. Mat. Univ. Parma, 5 (2014), 373-386.

[15]

A. Iannizzotto and M. Squassina, $\frac{1}{2}$ -Laplacian problems with exponential nonlinearity, J. Math. Anal. Appl, 414 (2014), 372-385. doi: 10.1016/j.jmaa.2013.12.059.

[16]

Yan Yan Li and Itai Shafrir, Blow-up analysis for solutions of -∆u = Veu in dimension two, Indiana Univ. Math. J, 43 (1994), 1255-1270. doi: 10.1512/iumj.1994.43.43054.

[17]

E. Di 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.

[18]

T. Ozawa, On critical cases of Sobolev's inequalities, J. Funct. Analysis, 127 (1995), 259-269. doi: 10.1006/jfan.1995.1012.

[19]

R. Servadei and E. Valdinoci, Mountain Pass solutions for non-local elliptic operators, J. Math. Anal. Appl, 389 (2012), 887-898. doi: 10.1016/j.jmaa.2011.12.032.

[20]

R. Servadei and E. Valdinoci, Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators, Rev. Mat. Iberoam, 29 (2013), 1091-1126. doi: 10.4171/RMI/750.

[21]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst, 33 (2013), 2105-2137.

[22]

Y. Sire and E. Valdinoci, Rigidity results for some boundary quasilinear phase transitions, Comm. Partial. Differ. Eq, 33 (2009), 765-784. doi: 10.1080/03605300902892402.

[23]

N. S. Trudinger, On imbeddings into Orcliz spaces and some applications, J. Math. Mech, 17 (1967), 473-483.

[24]

J. L. Vázquez, Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators, Discrete Contin. Dyn. Syst, 7 (2014), 857-885. doi: 10.3934/dcdss.2014.7.857.

[1]

Gang Cai, Yekini Shehu, Olaniyi S. Iyiola. Inertial Tseng's extragradient method for solving variational inequality problems of pseudo-monotone and non-Lipschitz operators. Journal of Industrial and Management Optimization, 2022, 18 (4) : 2873-2902. doi: 10.3934/jimo.2021095

[2]

Shaotao Hu, Yuanheng Wang, Bing Tan, Fenghui Wang. Inertial iterative method for solving variational inequality problems of pseudo-monotone operators and fixed point problems of nonexpansive mappings in Hilbert spaces. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022060

[3]

Zijia Peng, Cuiming Ma, Zhonghui Liu. Existence for a quasistatic variational-hemivariational inequality. Evolution Equations and Control Theory, 2020, 9 (4) : 1153-1165. doi: 10.3934/eect.2020058

[4]

Li Wang, Yang Li, Liwei Zhang. A differential equation method for solving box constrained variational inequality problems. Journal of Industrial and Management Optimization, 2011, 7 (1) : 183-198. doi: 10.3934/jimo.2011.7.183

[5]

Hui-Qiang Ma, Nan-Jing Huang. Neural network smoothing approximation method for stochastic variational inequality problems. Journal of Industrial and Management Optimization, 2015, 11 (2) : 645-660. doi: 10.3934/jimo.2015.11.645

[6]

Ming Chen, Chongchao Huang. A power penalty method for a class of linearly constrained variational inequality. Journal of Industrial and Management Optimization, 2018, 14 (4) : 1381-1396. doi: 10.3934/jimo.2018012

[7]

Walter Allegretto, Yanping Lin, Shuqing Ma. On the box method for a non-local parabolic variational inequality. Discrete and Continuous Dynamical Systems - B, 2001, 1 (1) : 71-88. doi: 10.3934/dcdsb.2001.1.71

[8]

Shuang Chen, Li-Ping Pang, Dan Li. An inexact semismooth Newton method for variational inequality with symmetric cone constraints. Journal of Industrial and Management Optimization, 2015, 11 (3) : 733-746. doi: 10.3934/jimo.2015.11.733

[9]

Yekini Shehu, Olaniyi Iyiola. On a modified extragradient method for variational inequality problem with application to industrial electricity production. Journal of Industrial and Management Optimization, 2019, 15 (1) : 319-342. doi: 10.3934/jimo.2018045

[10]

Chibueze Christian Okeke, Abdulmalik Usman Bello, Lateef Olakunle Jolaoso, Kingsley Chimuanya Ukandu. Inertial method for split null point problems with pseudomonotone variational inequality problems. Numerical Algebra, Control and Optimization, 2021  doi: 10.3934/naco.2021037

[11]

Yarui Duan, Pengcheng Wu, Yuying Zhou. Penalty approximation method for a double obstacle quasilinear parabolic variational inequality problem. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022017

[12]

Fouad Hadj Selem, Hiroaki Kikuchi, Juncheng Wei. Existence and uniqueness of singular solution to stationary Schrödinger equation with supercritical nonlinearity. Discrete and Continuous Dynamical Systems, 2013, 33 (10) : 4613-4626. doi: 10.3934/dcds.2013.33.4613

[13]

C. R. Chen, S. J. Li. Semicontinuity of the solution set map to a set-valued weak vector variational inequality. Journal of Industrial and Management Optimization, 2007, 3 (3) : 519-528. doi: 10.3934/jimo.2007.3.519

[14]

Abd-semii Oluwatosin-Enitan Owolabi, Timilehin Opeyemi Alakoya, Adeolu Taiwo, Oluwatosin Temitope Mewomo. A new inertial-projection algorithm for approximating common solution of variational inequality and fixed point problems of multivalued mappings. Numerical Algebra, Control and Optimization, 2022, 12 (2) : 255-278. doi: 10.3934/naco.2021004

[15]

Claudianor O. Alves, Geilson F. Germano. Existence of ground state solution and concentration of maxima for a class of indefinite variational problems. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2887-2906. doi: 10.3934/cpaa.2020126

[16]

Suxiang He, Pan Zhang, Xiao Hu, Rong Hu. A sample average approximation method based on a D-gap function for stochastic variational inequality problems. Journal of Industrial and Management Optimization, 2014, 10 (3) : 977-987. doi: 10.3934/jimo.2014.10.977

[17]

Francis Akutsah, Akindele Adebayo Mebawondu, Hammed Anuoluwapo Abass, Ojen Kumar Narain. A self adaptive method for solving a class of bilevel variational inequalities with split variational inequality and composed fixed point problem constraints in Hilbert spaces. Numerical Algebra, Control and Optimization, 2021  doi: 10.3934/naco.2021046

[18]

Wenjun Liu, Jiangyong Yu, Gang Li. Global existence, exponential decay and blow-up of solutions for a class of fractional pseudo-parabolic equations with logarithmic nonlinearity. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4337-4366. doi: 10.3934/dcdss.2021121

[19]

Takeshi Fukao. Variational inequality for the Stokes equations with constraint. Conference Publications, 2011, 2011 (Special) : 437-446. doi: 10.3934/proc.2011.2011.437

[20]

Augusto VisintiN. On the variational representation of monotone operators. Discrete and Continuous Dynamical Systems - S, 2017, 10 (4) : 909-918. doi: 10.3934/dcdss.2017046

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (266)
  • HTML views (140)
  • Cited by (9)

Other articles
by authors

[Back to Top]