Topological Methods in Nonlinear Analysis
- Topol. Methods Nonlinear Anal.
- Volume 48, Number 2 (2016), 477-492.
Ground states of nonlocal scalar field equations with Trudinger-Moser critical nonlinearity
João Marcos do Ó, Olímpio H. Miyagaki, and Marco Squassina
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Abstract
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.
Article information
Source
Topol. Methods Nonlinear Anal., Volume 48, Number 2 (2016), 477-492.
Dates
First available in Project Euclid: 21 December 2016
Permanent link to this document
https://projecteuclid.org/euclid.tmna/1482289225
Digital Object Identifier
doi:10.12775/TMNA.2016.045
Mathematical Reviews number (MathSciNet)
MR3642769
Zentralblatt MATH identifier
1375.35184
Citation
do Ó, João Marcos; Miyagaki, Olímpio H.; Squassina, Marco. Ground states of nonlocal scalar field equations with Trudinger-Moser critical nonlinearity. Topol. Methods Nonlinear Anal. 48 (2016), no. 2, 477--492. doi:10.12775/TMNA.2016.045. https://projecteuclid.org/euclid.tmna/1482289225
References
- Adimurthi and S.L. Yadava, Multiplicity results for semilinear elliptic equations in a bounded domain of $\R^2$ 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 $\mathbb{R}^2$ 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 $\mathbb{R}^2$, 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.
- ––––, 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 $\R^2$ 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.
- ––––, 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'skiĭ and J.B. Rutickiĭ, 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 $\R^2$ 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 $\R^N$ with critical growth, J. Math. Phys. 54 (2013), 121502-19 pages.
- S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in $\R^N$, J. Math. Phys. 54 (2013), 031501-17 pages. Mathematical Reviews (MathSciNet): MR3059423
Zentralblatt MATH: 1281.81034
Digital Object Identifier: doi:10.1063/1.4793990 - 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.
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