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June  2018, 11(3): 561-576. doi: 10.3934/dcdss.2018031

Bifurcation results for problems with fractional Trudinger-Moser nonlinearity

Kanishka Perera 1, and  Marco Squassina 2,

1. 

Department of Mathematical Sciences, Florida Institute of Technology, 150 W University Blvd, Melbourne, FL 32901, USA
2. 

Dipartimento di Matematica e Fisica, Università Cattolica del Sacro Cuore, Via dei Musei 41, I-25121 Brescia, Italy

Received  May 2017 Revised  August 2017 Published  October 2017

Fund Project: The second author is member of Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

By using a suitable topological argument based on cohomological linking and by exploiting a Trudinger-Moser inequality in fractional spaces recently obtained, we prove existence of multiple solutions for a problem involving the nonlinear fractional laplacian and a related critical exponential nonlinearity. This extends the literature for the $N$-Laplacian operator.

Keywords: Fractional Trudinger-Moser embedding, exponential nonlinearity, existence of solutions.
Mathematics Subject Classification: Primary: 35J92; Secondary: 35P30.
Citation: Kanishka Perera, Marco Squassina. Bifurcation results for problems with fractional Trudinger-Moser nonlinearity. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 561-576. doi: 10.3934/dcdss.2018031
References:
[1]

Adimurthi, Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the $n$-Laplacian, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 17 (1990), 393-413.   Google Scholar

[2]

Adimurthi and S. L. Yadava, Bifurcation results for semilinear elliptic problems with critical exponent in $\mathbb R^2$, Nonlinear Anal., 14 (1990), 607-612.  doi: 10.1016/0362-546X(90)90065-O.  Google Scholar

[3]

P. Bartolo, V. Benci and D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity, Nonlinear Anal., 7 (1983), 981-1012.  doi: 10.1016/0362-546X(83)90115-3.  Google Scholar

[4]

V. Benci, On critical point theory for indefinite functionals in the presence of symmetries, Trans. Amer. Math. Soc., 274 (1982), 533-572.  doi: 10.1090/S0002-9947-1982-0675067-X.  Google Scholar

[5]

L. Brasco, E. Parini and M. Squassina, Stability of variational eigenvalues for the fractional $p$-Laplacian, Discrete Contin. Dyn. Syst. A, 36 (2016), 1813-1845.  doi: 10.3934/dcds.2016.36.1813.  Google Scholar

[6]

H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.  doi: 10.2307/2044999.  Google Scholar

[7]

L. Carleson and A. Chang, On the existence of an extremal function for an inequality of J. Moser, Bull. Sci. Math., 110 (1986), 113-127.   Google Scholar

[8]

D. G. de Figueiredo, O. H. Miyagaki and B. Ruf, Elliptic equations in $\mathbb R^2$ with nonlinearities in the critical growth range, Calc. Var. Partial Differential Equations, 3 (1995), 139-153.  doi: 10.1007/BF01205003.  Google Scholar

[9]

D. G. de Figueiredo, O. H. Miyagaki and B. Ruf, Corrigendum to Elliptic equations in $\mathbb R^2$ with nonlinearities in the critical growth range Calc. Var. Partial Differential Equations 4 (1996), p203. doi: 10.1007/BF01189953.  Google Scholar

[10]

J. M. do'O, Semilinear dirichlet problems for the $N$-Laplacian in $\mathbb R^N$ with nonlinearities in the critical growth range, Differential Integral Equations, 9 (1996), 967-979.   Google Scholar

[11]

J. M. do'O, O. H. Miyagaki and M. Squassina, Nonautonomous fractional problems with exponential growth, NoDEA Nonlinear Differential Equations Applications, 22 (2015), 1395-1410.  doi: 10.1007/s00030-015-0327-0.  Google Scholar

[12]

J. M. do'O, O. H. Miyagaki and M. Squassina, Ground states of nonlocal scalar field equations with Trudinger-Moser critical nonlinearity, Topol. Meth. Nonlinear Anal., 48 (2016), 477-492.   Google Scholar

[13]

E. R. Fadell and P. H. Rabinowitz, Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems, Invent. Math., 45 (1978), 139-174.  doi: 10.1007/BF01390270.  Google Scholar

[14]

J. Giacomoni, P. K. Mishra and K. Sreenadh, Fractional elliptic equations with critical exponential nonlinearity, Adv. Nonlinear Anal., 5 (2016), 57-74.  doi: 10.1515/anona-2015-0081.  Google Scholar

[15]

A. Iannizzotto, S. Mosconi and M. Squassina, Global Holder regularity for the fractional $p$-Laplacian, Rev. Mat. Iberoam., 32 (2016), 1355-1394.  doi: 10.4171/RMI/921.  Google Scholar

[16]

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

[17]

S. Iula, A note on the Moser-Trudinger inequality in Sobolev-Slobodeckij spaces in dimension one, , (2016).   Google Scholar

[18]

S. Iula, A. Maalaoui and L. Martinazzi, A fractional Moser-Trudinger type inequality in one dimension and its critical points, Differential Integral Equations, 29 (2016), 455-492.   Google Scholar

[19]

H. Kozono, T. Sato and H. Wadade, Upper bound of the best constant of a Trudinger-Moser inequality and its application to a Gagliardo-Nirenberg inequality, Indiana Univ. Math. J., 55 (2006), 1951-1974.  doi: 10.1512/iumj.2006.55.2743.  Google Scholar

[20]

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case I, Rev. Mat. Iberoam., 1 (1985), 145-201.  doi: 10.4171/RMI/6.  Google Scholar

[21]

L. Martinazzi, Fractional Adams-Moser-Trudinger type inequalities, Nonlinear Anal., 127 (2015), 263-278.  doi: 10.1016/j.na.2015.06.034.  Google Scholar

[22]

S. Mosconi and M. Squassina, Recent progresses in the theory of nonlinear nonlocal problems, Bruno Pini Math. Analysis Sem., 2016 (2016), 147-164.   Google Scholar

[23]

J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20 (1970/71), 1077-1092.  doi: 10.1512/iumj.1971.20.20101.  Google Scholar

[24]

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

[25]

E. Parini and B. Ruf, On the Moser-Trudinger inequality in fractional Sobolev-Slobodeckij spaces, J. Anal. Math., to appear, arXiv: 1607.07681. Google Scholar

[26]

K. Perera, R. P. Agarwal and D. O'Regan, Morse Theoretic Aspects of $p$-Laplacian Type Operators 161, Morse Theoretic Aspects of p{Laplacian Type Operators 161, Mathematical surveys and monographs, American Mathematical Society, Providence, RI, (2010).  doi: 10.1090/surv/161.  Google Scholar

[27]

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

[28]

Y. Yang and K. Perera, $N$-Laplacian problems with critical Trudinger-Moser nonlinearities, Ann. Sc. Norm. Super. Pisa Cl. Sci., 16 (2016), 1123-1138.   Google Scholar

show all references

References:
[1]

Adimurthi, Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the $n$-Laplacian, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 17 (1990), 393-413.   Google Scholar

[2]

Adimurthi and S. L. Yadava, Bifurcation results for semilinear elliptic problems with critical exponent in $\mathbb R^2$, Nonlinear Anal., 14 (1990), 607-612.  doi: 10.1016/0362-546X(90)90065-O.  Google Scholar

[3]

P. Bartolo, V. Benci and D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity, Nonlinear Anal., 7 (1983), 981-1012.  doi: 10.1016/0362-546X(83)90115-3.  Google Scholar

[4]

V. Benci, On critical point theory for indefinite functionals in the presence of symmetries, Trans. Amer. Math. Soc., 274 (1982), 533-572.  doi: 10.1090/S0002-9947-1982-0675067-X.  Google Scholar

[5]

L. Brasco, E. Parini and M. Squassina, Stability of variational eigenvalues for the fractional $p$-Laplacian, Discrete Contin. Dyn. Syst. A, 36 (2016), 1813-1845.  doi: 10.3934/dcds.2016.36.1813.  Google Scholar

[6]

H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.  doi: 10.2307/2044999.  Google Scholar

[7]

L. Carleson and A. Chang, On the existence of an extremal function for an inequality of J. Moser, Bull. Sci. Math., 110 (1986), 113-127.   Google Scholar

[8]

D. G. de Figueiredo, O. H. Miyagaki and B. Ruf, Elliptic equations in $\mathbb R^2$ with nonlinearities in the critical growth range, Calc. Var. Partial Differential Equations, 3 (1995), 139-153.  doi: 10.1007/BF01205003.  Google Scholar

[9]

D. G. de Figueiredo, O. H. Miyagaki and B. Ruf, Corrigendum to Elliptic equations in $\mathbb R^2$ with nonlinearities in the critical growth range Calc. Var. Partial Differential Equations 4 (1996), p203. doi: 10.1007/BF01189953.  Google Scholar

[10]

J. M. do'O, Semilinear dirichlet problems for the $N$-Laplacian in $\mathbb R^N$ with nonlinearities in the critical growth range, Differential Integral Equations, 9 (1996), 967-979.   Google Scholar

[11]

J. M. do'O, O. H. Miyagaki and M. Squassina, Nonautonomous fractional problems with exponential growth, NoDEA Nonlinear Differential Equations Applications, 22 (2015), 1395-1410.  doi: 10.1007/s00030-015-0327-0.  Google Scholar

[12]

J. M. do'O, O. H. Miyagaki and M. Squassina, Ground states of nonlocal scalar field equations with Trudinger-Moser critical nonlinearity, Topol. Meth. Nonlinear Anal., 48 (2016), 477-492.   Google Scholar

[13]

E. R. Fadell and P. H. Rabinowitz, Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems, Invent. Math., 45 (1978), 139-174.  doi: 10.1007/BF01390270.  Google Scholar

[14]

J. Giacomoni, P. K. Mishra and K. Sreenadh, Fractional elliptic equations with critical exponential nonlinearity, Adv. Nonlinear Anal., 5 (2016), 57-74.  doi: 10.1515/anona-2015-0081.  Google Scholar

[15]

A. Iannizzotto, S. Mosconi and M. Squassina, Global Holder regularity for the fractional $p$-Laplacian, Rev. Mat. Iberoam., 32 (2016), 1355-1394.  doi: 10.4171/RMI/921.  Google Scholar

[16]

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

[17]

S. Iula, A note on the Moser-Trudinger inequality in Sobolev-Slobodeckij spaces in dimension one, , (2016).   Google Scholar

[18]

S. Iula, A. Maalaoui and L. Martinazzi, A fractional Moser-Trudinger type inequality in one dimension and its critical points, Differential Integral Equations, 29 (2016), 455-492.   Google Scholar

[19]

H. Kozono, T. Sato and H. Wadade, Upper bound of the best constant of a Trudinger-Moser inequality and its application to a Gagliardo-Nirenberg inequality, Indiana Univ. Math. J., 55 (2006), 1951-1974.  doi: 10.1512/iumj.2006.55.2743.  Google Scholar

[20]

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case I, Rev. Mat. Iberoam., 1 (1985), 145-201.  doi: 10.4171/RMI/6.  Google Scholar

[21]

L. Martinazzi, Fractional Adams-Moser-Trudinger type inequalities, Nonlinear Anal., 127 (2015), 263-278.  doi: 10.1016/j.na.2015.06.034.  Google Scholar

[22]

S. Mosconi and M. Squassina, Recent progresses in the theory of nonlinear nonlocal problems, Bruno Pini Math. Analysis Sem., 2016 (2016), 147-164.   Google Scholar

[23]

J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20 (1970/71), 1077-1092.  doi: 10.1512/iumj.1971.20.20101.  Google Scholar

[24]

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

[25]

E. Parini and B. Ruf, On the Moser-Trudinger inequality in fractional Sobolev-Slobodeckij spaces, J. Anal. Math., to appear, arXiv: 1607.07681. Google Scholar

[26]

K. Perera, R. P. Agarwal and D. O'Regan, Morse Theoretic Aspects of $p$-Laplacian Type Operators 161, Morse Theoretic Aspects of p{Laplacian Type Operators 161, Mathematical surveys and monographs, American Mathematical Society, Providence, RI, (2010).  doi: 10.1090/surv/161.  Google Scholar

[27]

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

[28]

Y. Yang and K. Perera, $N$-Laplacian problems with critical Trudinger-Moser nonlinearities, Ann. Sc. Norm. Super. Pisa Cl. Sci., 16 (2016), 1123-1138.   Google Scholar

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