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Optimal elliptic regularity: A comparison between local and nonlocal equations
Bifurcation results for problems with fractional Trudinger-Moser nonlinearity
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 |
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.
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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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.
|
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[17] |
S. Iula,
A note on the Moser-Trudinger inequality in Sobolev-Slobodeckij spaces in dimension one, , (2016).
|
[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.
|
[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. |
[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. |
[21] |
L. Martinazzi,
Fractional Adams-Moser-Trudinger type inequalities, Nonlinear Anal., 127 (2015), 263-278.
doi: 10.1016/j.na.2015.06.034. |
[22] |
S. Mosconi and M. Squassina,
Recent progresses in the theory of nonlinear nonlocal problems, Bruno Pini Math. Analysis Sem., 2016 (2016), 147-164.
|
[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. |
[24] |
T. Ozawa,
On critical cases of Sobolev's inequalitites, J. Funct. Anal., 127 (1995), 259-269.
doi: 10.1006/jfan.1995.1012. |
[25] |
E. Parini and B. Ruf, On the Moser-Trudinger inequality in fractional Sobolev-Slobodeckij
spaces, J. Anal. Math., to appear, arXiv: 1607.07681. |
[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. |
[27] |
N. S. Trudinger,
On imbeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-483.
|
[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.
|
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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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.
|
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[17] |
S. Iula,
A note on the Moser-Trudinger inequality in Sobolev-Slobodeckij spaces in dimension one, , (2016).
|
[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.
|
[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. |
[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. |
[21] |
L. Martinazzi,
Fractional Adams-Moser-Trudinger type inequalities, Nonlinear Anal., 127 (2015), 263-278.
doi: 10.1016/j.na.2015.06.034. |
[22] |
S. Mosconi and M. Squassina,
Recent progresses in the theory of nonlinear nonlocal problems, Bruno Pini Math. Analysis Sem., 2016 (2016), 147-164.
|
[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. |
[24] |
T. Ozawa,
On critical cases of Sobolev's inequalitites, J. Funct. Anal., 127 (1995), 259-269.
doi: 10.1006/jfan.1995.1012. |
[25] |
E. Parini and B. Ruf, On the Moser-Trudinger inequality in fractional Sobolev-Slobodeckij
spaces, J. Anal. Math., to appear, arXiv: 1607.07681. |
[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. |
[27] |
N. S. Trudinger,
On imbeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-483.
|
[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.
|
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