# American Institute of Mathematical Sciences

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 & 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [4] G. Molica Bisci, V. Radulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Combridge University press, Cambridge, 2015. doi: 10.1017/CBO9781316282397.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [9] I. Ekeland, On the variational principle principle, J. Math. Anal. Appl, 47 (2014), 324-353.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [13] A. Fiscella, R. Servadei and E. Valdinoci, Density properties for fractional Sobolev spaces, preprint. doi: 10.5186/aasfm.2015.4009.  Google Scholar [14] G. Franzina and G. Palatucci, Fractional p-eigenvalues, Riv. Mat. Univ. Parma, 5 (2014), 373-386.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] T. Ozawa, On critical cases of Sobolev's inequalities, J. Funct. Analysis, 127 (1995), 259-269. doi: 10.1006/jfan.1995.1012.  Google Scholar [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.  Google Scholar [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.  Google Scholar [21] R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst, 33 (2013), 2105-2137.  Google Scholar [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.  Google Scholar [23] N. S. Trudinger, On imbeddings into Orcliz spaces and some applications, J. Math. Mech, 17 (1967), 473-483.  Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [4] G. Molica Bisci, V. Radulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Combridge University press, Cambridge, 2015. doi: 10.1017/CBO9781316282397.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [9] I. Ekeland, On the variational principle principle, J. Math. Anal. Appl, 47 (2014), 324-353.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [13] A. Fiscella, R. Servadei and E. Valdinoci, Density properties for fractional Sobolev spaces, preprint. doi: 10.5186/aasfm.2015.4009.  Google Scholar [14] G. Franzina and G. Palatucci, Fractional p-eigenvalues, Riv. Mat. Univ. Parma, 5 (2014), 373-386.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] T. Ozawa, On critical cases of Sobolev's inequalities, J. Funct. Analysis, 127 (1995), 259-269. doi: 10.1006/jfan.1995.1012.  Google Scholar [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.  Google Scholar [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.  Google Scholar [21] R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst, 33 (2013), 2105-2137.  Google Scholar [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.  Google Scholar [23] N. S. Trudinger, On imbeddings into Orcliz spaces and some applications, J. Math. Mech, 17 (1967), 473-483.  Google Scholar [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.  Google Scholar
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