Advanced Search
Article Contents
Article Contents

Bifurcation results for problems with fractional Trudinger-Moser nonlinearity

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).
Abstract Full Text(HTML) Related Papers Cited by
  • 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.

    Mathematics Subject Classification: Primary: 35J92; Secondary: 35P30.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [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. BartoloV. 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. BrascoE. 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 FigueiredoO. 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'OO. 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'OO. 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. GiacomoniP. 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. IannizzottoS. 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. IulaA. 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. KozonoT. 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. PereraR. P. Agarwal and D. O'ReganMorse 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. 
  • 加载中

Article Metrics

HTML views(1959) PDF downloads(179) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint