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.

[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]

M. Giovanni 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]

Brezis Haim and Merle Frank, 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. CafarelliS. 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. CaffarelliJ-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. FelmerA. 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 FigueiredoJ. M. do Ò and B. Ruf, On an inequality by Trudinger N., J. Moser and related elliptic equations, Commun. Pure Appl. Math, 55 (2002), 135-152. doi: 10.1002/cpa.10015.

[12]

D. G. de FigueiredoO. 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 Li Yan and Shafrir Itai, 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 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.

[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]

M. Giovanni 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]

Brezis Haim and Merle Frank, 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. CafarelliS. 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. CaffarelliJ-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. FelmerA. 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 FigueiredoJ. M. do Ò and B. Ruf, On an inequality by Trudinger N., J. Moser and related elliptic equations, Commun. Pure Appl. Math, 55 (2002), 135-152. doi: 10.1002/cpa.10015.

[12]

D. G. de FigueiredoO. 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 Li Yan and Shafrir Itai, 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 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.

[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.

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