Concentration-compactness for singular nonlocal Schrödinger equations with oscillatory nonlinearities

João Marcos do Ó, Diego Ferraz



The paper is dedicated to the theory of concentration-compactness principles for inhomogeneous fractional Sobolev spaces. This subject for the local case has been studied since the publication of the celebrated works due to P.-L. Lions, which laid the broad foundations of the method and outlined a wide scope of its applications. Our study is based on the analysis of the profile decomposition for the weak convergence following the approach of dislocation spaces, introduced by K. Tintarev and \hbox{K.-H. Fieseler}. As an application, we obtain existence of nontrivial and nonnegative solutions and ground states for fractional Schrödinger equations for a wide class of possible singular potentials, not necessarily bounded away from zero. We consider possible oscillatory nonlinearities for both cases, subcritical and critical which are superlinear at the origin, without the classical Ambrosetti and Rabinowitz growth condition. In some of our results we prove existence of solutions by means of compactness of Palais-Smale sequences of the associated functional at the mountain pass level. To this end we study and provide the behavior of the weak profile decomposition convergence under the related functionals. Moreover, we use a Pohozaev type identity in our argument to compare the minimax levels of the energy functional with the ones of the associated limit problem. Motivated by this fact, in our work we also prove that this kind of identities hold for a larger class of potentials and nonlinearities for the fractional framework.


Fractional Schrödinger equation; concentration-compactness principle; critical Sobolev exponents

Full Text:



A. Ambrosetti, V. Felli and A. Malchiodi, Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Eur. Math. Soc. (JEMS) 7 (2005), 117–144.

D. Applebaum, Lévy processes – from probability to finance and quantum groups, Notices Amer. Math. Soc. 51 (2004), 1336–1347.

W. Beckner, Sobolev inequalities, the Poisson semigroup, and analysis on the sphere S n , Proc. Natl. Acad. Sci. USA 89 (1992), 4816–4819.

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), 1245–1260.

X. Chang, Ground states of some fractional Schrödinger equations on RN , Proc. Edinb. Math. Soc. (2) 58 (2015), 305–321.

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.G. Costa, J.M. do Ó and K. Tintarev, Schrödinger equations with critical nonlinearity, singular potential and a ground state, J. Differential Equations 249 (2010), 240–252.

M. Cwikel and K. Tintarev, On interpolation of cocompact imbeddings, Rev. Mat. Complut. 26 (2013), 33–55.

R. de Marchi, Schrödinger equations with asymptotically periodic terms, Proc. Roy. Soc. Edinburgh Sect. A 145 (2015), 745–757.

M. de Souza, J.M. do Ó and T. da Silva, On a class quasilinear Schrödinger equations in Rn , Appl. Anal. 95 (2016), 323–340.

Y. Deng, L. Jin and S. Peng, Solutions of Schrödinger equations with inverse square potential and critical nonlinearity, J. Differential Equations 253 (2012), 1376–1398.

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

Y. Ding and C. Lee, Multiple solutions of Schrödinger equations with indefinite linear part and super or asymptotically linear terms, J. Differential Equations 222 (2006), 137–163.

S. Dipierro, L. Montoro, I. Peral and B. Sciunzi, Qualitative properties of positive solutions to nonlocal critical problems involving the Hardy–Leray potential,

S. Dipierro, G. Palatucci and E. Valdinoci, Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian, Matematiche (Catania) 68 (2013), 201–216.

J.M. do Ó and D. Ferraz, Concentration-compactness principle for nonlocal scalar field equations with critical growth, J. Math. Anal. Appl. (2016),

M.M. Fall and V. Felli, Unique continuation property and local asymptotics of solutions to fractional elliptic equations, Comm. Partial Differential Equations 39 (2014), 354–397.

M.M. Fall and T. Weth, Monotonicity and nonexistence results for some fractional elliptic problems in the half-space, Commun. Contemp. Math. 18 (2016), 1550012, 25.

V. Felli and A. Pistoia, Existence of blowing-up solutions for a nonlinear elliptic equation with Hardy potential and critical growth, Comm. Partial Differential Equations 31 (2006), 21–56.

V. Felli and S. Terracini, Elliptic equations with multi-singular inverse-square potentials and critical nonlinearity, Comm. Partial Differential Equations 31 (2006), 469–495.

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

B. Feng, Ground states for the fractional Schrödinger equation, Electron. J. Differential Equations (2013), No. 127, 11.

P. Gérard, Description du défaut de compacité de l’injection de Sobolev, ESAIM Control Optim. Calc. Var. 3 (1998), 213–233.

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. (JEMS) 16 (2014), 1111–1171.

R. Lehrer, L.A. Maia and M. Squassina, Asymptotically linear fractional Schrödinger equations, Complex Var. Elliptic Equ. 60 (2015), 529–558.

S. Li, Y. Ding and Y. Chen, Concentrating standing waves for the fractional Schrödinger equation with critical nonlinearities, Bound. Value Probl. (2015), 2015:240.

H.F. Lins and E.A.B. Silva, Quasilinear asymptotically periodic elliptic equations with critical growth, Nonlinear Anal. 71 (2009), 2890–2905.

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case I, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), 109–145.

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case II, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), 223–283.

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case I, Rev. Mat. Iberoamericana 1 (1985), 145–201.

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case II, Rev. Mat. Iberoamericana 1 (1985), 45–121.

O. H. Miyagaki, On a class of semilinear elliptic problems in RN with critical growth, Nonlinear Anal. 29 (1997), 773–781.

A. Nekvinda, Characterization of traces of the weighted Sobolev space W 1,p (Ω, dM ) on M , Czechoslovak Math. J. 43 (1993), no. 118, 695–711.

G. Palatucci and A. Pisante, Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces, Calc. Var. Partial Differential Equations 50 (2014), 799–829.

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

X. Ros-Oton and J. Serra, The Pohozaev identity for the fractional Laplacian, Arch. Ration. Mech. Anal. 213 (2014), 587–628.

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

X. Shang and J. Zhang, Ground states for fractional Schrödinger equations with critical growth, Nonlinearity 27 (2014), 187–207.

X. Shang, J. Zhang and Y. Yang, On fractional Schrödinger equation in RN with critical growth, J. Math. Phys. 54 (2013), 121502, 20.

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math. 60 (2007), 67–112.

B. Sirakov, Existence and multiplicity of solutions of semi-linear elliptic equations in RN , Calc. Var. Partial Differential Equations 11 (2000), 119–142.

D. Smets, Nonlinear Schrödinger equations with Hardy potential and critical nonlinearities, Trans. Amer. Math. Soc. 357 (2005), 2909–2938.

S. Solimini, A note on compactness-type properties with respect to Lorentz norms of bounded subsets of a Sobolev space, Ann. Inst. H. Poincaré Anal. Non Linéaire 12 (1995), 319–337.

M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z. 187 (1984), 511–517.

A. Szulkin and T. Weth, The method of Nehari manifold, Handbook of Nonconvex Analysis and Applications, Int. Press, Somerville, MA, 2010, 597–632.

S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent, Adv. Differential Equations 1 (1996), 241–264.

K. Tintarev, Positive solutions of elliptic equations with a critical oscillatory nonlinearity, Discrete Contin. Dyn. Syst. (2007), 974–981.

K. Tintarev, Concentration compactness at the mountain pass level in semilinear elliptic problems, NoDEA Nonlinear Differential Equations Appl. 15 (2008), 581–598.

K. Tintarev and K.-H. Fieseler, Concentration compactness, Functional-Analytic Grounds and Applications, Imperial College Press, London, 2007.

D. Yafaev, Sharp constants in the Hardy–Rellich inequalities, J. Funct. Anal. 168 (1999), 121–144.

H. Zhang, J. Xu and F. Zhang, Existence and multiplicity of solutions for superlinear fractional Schrödinger equations in RN , J. Math. Phys. 56 (2015), 091502, 13.


  • There are currently no refbacks.

Partnerzy platformy czasopism