Ground states of nonlocal scalar field equations with Trudinger-Moser critical nonlinearity

João Marcos do Ó, Olímpio H. Miyagaki, Marco Squassina



We investigate the existence of ground state solutions for a class of nonlinear scalar field equations defined on the whole real line, involving a fractional Laplacian and nonlinearities with Trudinger-Moser critical growth. We handle the lack of compactness of the associated energy functional due to the unboundedness of the domain and the presence of a limiting case embedding.


Trudinger-Moser inequality; fractional Laplacian; ground states

Full Text:



Adimurthi and S.L. Yadava, Multiplicity results for semilinear elliptic equations in a bounded domain of R2 involving critical exponent, Ann. Sc. Norm. Super. Pisa Cl. Sci. (1990), 481–504.

C. Alves, J. M. do Ó and O. Miyagaki, On nonlinear perturbations of a periodic elliptic problem in R2 involving critical growth, Nonlinear Anal. 56 (2004), 781–791.

C. Alves, M. Souto and M. Montenegro, Existence of a ground state solution for a nonlinear scalar field equation with critical growth, Calc. Var. Partial Differential Equations 43 (2012), 537–554.

A. Ambrosetti and Z.-Q. Wang, Positive solutions to a class of quasilinear elliptic equations on R, Discrete Contin. Dyn. Syst. 9 (2003), 55–68.

B. Barrios, E. Colorado, A. de Pablo and U. Sánchez, On some critical problems for the fractional Laplacian operator, J. Differential Equations 252 (2012), 613–6162.

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations, I-existence of a ground state, Arch. Rat. Mech. Anal. 82 (1983), 313–346.

H. Berestycki, T. Gallouet and O. Kavian, Equations de Champs scalaires euclidiens non lineaires dans le plan, C.R. Acad. Sci. Paris Sér. I Math. 297 (1984), 307–310.

H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), 486–490.

X. Cabré and J.G. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math. 224 (2010), 2052–2093.

X. Chang and Z-Q. Wang, Ground state of scalar field equations involving a fractional laplacian with general nonlinearity, Nonlinearity 26 (2013), 479–494.

D. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent in R2, Comm. Partial Differential Equations 17 (1992), 407–435.

D.G. de Figueiredo, J.M. do Ó and B. Ruf, Elliptic equations and systems with critical Trudinge-r-Moser nonlinearities, Discrete Contin. Dyn. Syst. 30 (2011), 455–476.

D.G. de Figueiredo, J.M. do Ó and B. Ruf, On an inequality by N. Trudinger and J. Moser and related elliptic equations, Comm. Pure Appl. Math. 55 (2002), 135–152.

D.G. de Figueiredo, O.H. Miyagaki and B. Ruf, Elliptic equations in R2 with nonlinearities in the critical growth range, Calc. Var. Partial Differential Equations 3 (1995), 139–153.

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), 521–573.

J.M. do Ó, E. Medeiros and U. Severo, A nonhomogeneous elliptic problem involving critical growth in dimension two, J. Math. Anal. Appl. 345 (2008), 286–304.

J.M. do Ó, O.H. Miyagaki and M. Squassina, Critical and subcritical fractional problems with vanishing potentials, Comm. Contemporary Math., DOI: 10.1142/S0219199715500637.

J.M. do Ó, O.H. Miyagaki and M. Squassina, Nonautonomous fractional problems with exponential growth, Nonlinear Differential Equations Applications, DOI: 10.1007/s00030-015-0327-0.

P. Felmer, A. Quaas and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh A 142 (2012), 1237-1262.

R. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional laplacians in R, Acta Math. 210 (2013), 261–318.

A. Iannizzotto and M. Squassina, 1/2-Laplacian problems with exponential nonlinearity, J. Math. Anal. Appl. 414 (2014), 372–385.

T. Jin, Y. Li and J. Xiong, On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions, J. Eur. Math. Soc. 16 (2014), 1111–1171.

M.A. Krasnosel’skiı̆ and J.B. Rutickiı̆, ConvexFfunctions and Orlicz Spaces, Noordhoff, Groningen, Holland, 1961.

N. Lam and G. Lu, Elliptic equations and systems with subcritical and critical exponential growth without the Ambrosetti–Rabinowitz condition, J. Geom. Anal. 24 (2014), 118–143.

E. H. Lieb and M. Loss, Analysis, AMS, 2001.

J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1970), 1077–1092.

T. Ozawa, On critical cases of Sobolev’s inequalities, J. Funct. Anal. 127 (1995), 259–269.

P.H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys. 43 (1992), 270–291.

B. Ruf and F. Sani, Ground states for elliptic equations in R2 with exponential critical growth, Geometric properties for parabolic and elliptic PDE’s, Springer INdAM 2, Springer, Milan, 2013, 251–267.

X. Shang, J. Zhang and Y. Yang, On fractional Schödinger equation in RN with critical growth, J. Math. Phys. 54 (2013), 121502-19 pages.

S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in RN, J. Math. Phys. 54 (2013), 031501-17 pages.

M. Souza and Y. Araúujo, On nonlinear perturbations of a periodic fractional Schrödinger equation with critical exponential growth, Math. Nachr., to appear.

W. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977), 149–162.

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

M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24, Birkhäuser 1996.


  • There are currently no refbacks.

Partnerzy platformy czasopism