Multiple positive solutions for a fractional elliptic systems involving sign-changing weight

Haining Fan


We study multiplicity results for positive solutions for a fractional elliptic system involving both concave-convex and critical growth terms. With the help of Nehari manifold and Ljusternik-Schnirelmann category, we investigate how the coefficient $h$ of the critical nonlinearity affects the number of positive solutions to this problem and get a relationship between the number of positive solutions and the topology of the global maximum set of $h$.


Nehari manifold; fractional elliptic system; Lusternik-Schnirelmann category; multiple positive solutions

Full Text:



G. Alberti, G. Bouchitte and P. Seppecher, Phase transition with the line-tension effect, Arch. Rational Mech. Anal. 144 (1998), 1–46.

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

B. Barrios, E. Colorado, A. de Pablo and U. Sánchez, On some critical problems for the fractional Laplacian, J. Differential Equations 252 (2012), 6133–6162.

P.W. Bates, On some nonlocal evolution equations arising in materials science, Nonlinear Dynamics and Evolution Equations 48 (2006), 13–52.

L. Caffarelli, J.M. Roquejoffre and O. Savin, Nonlical minimal surfaces, Comm. Pure Apple. Math. 63 (2012), 1111–1144.

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

L. Caffarelli and E. Valdinoci, Uniform estimates and limiting arguments for nonlocal minimal surfaces, Calc. Var. Partial Differential Equations 32 (2007), 1245–1260.

S.Y. Chang and M. Gonzalez, Fractional Laplacian in conformal geometry, Adv. Math. 226 (2011), 1410–1432.

W. Chen and S. Deng, The Nehari manifold for nonlocal elliptic operators involving concave-convex nonlinearities, Z. Angew. Math. Phys. 66 (2015), 1387–1400.

W. Chen and S. Deng, The Nehari manifold for a fractional p-Laplacian system involving concave convex nonlinearities, Nonlinear Anal. 27 (2016), 80–92.

C.Y. Chen and T.F. Wu, The Nehari manifold for indefinite semilinear elliptic systems involving critical exponent, Appl. Math. Comp. 218 (2012), 10817–10828.

S. Cingolani and M. Lazzo, Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions, J. Differential Equations 160 (2000), 118–138.

R. Cont and P. Tankov, Financial modeling with jump processes, Chapman Hall/CRC Financial Mathematics Series, CRC Press, Boca Raton, 2004.

J. Dávila, M.del Pino and J.C. Wei, Concentrating standing waves for the fractional nonlinear Schrödinger equation, J. Differential Equations 256 (2014), 858–892.

G.M. Figueiredo and G. Siciliano, A multiplicity result via Lusternick–Schnirelmann category and Morse theory for a fractional Schrödinger equation in RN , arXiv: 1502. 01243v1 2015.

S. Goyal and K. Sreenadh, A Nehari manifold for non-local elliptic operator with concave-convex non-linearities and sign-changing weight function, arXiv:1307.5149, 2016.

X. He, M. Squassina and W. Zou, The Nehari manifold for fractional systems involving critical nonlinearities, arXiv:1509.02713, 2016.

R. Metzler and J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamic approach, Phys. Rep. 339 (2000), 63–77.

R. Metzler and J. Klafter, The restaurant at the random walk: recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A 37 (2004), 161–208.

E. Milakis and L. Silvestre, Regularity for the nonlinear Signorini problem, Adv. Math. 217 (2008), 1301–1312.

G. Palatucci and A. Pisante, Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces, Calc. Var., DOI: 10.1007/s00526013-0656-y.

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

X. Shang and J. Zhang, Concentrating solutions of nonlinear fractional Schrödinger equation with potentials, J. Differential Equations 258 (2015), 1106–1128.

X. Shang, J. Zhang and Y. Yang, On fractional Schrödinger equations in RN with critical growth, J. Math. Phys. 54 (2013), 121502–19.

X. Shang, J. Zhang and Y. Yang, Positive solutions of nonhomogeneous fractional Laplacian problem with critical exponent, Comm. Pure Appl. Anal. 13 (2014), 567–584.

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Apll. Math. 13 (1960), 457–468.

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

Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetric result, J. Funct. Anal. 60 (2007), 67–112.

K. Teng and X. He, Ground state solutions for fractional Schrödinger equations with critical Sobolev exponent, Commun. Pure Appl. Anal. 15 (2016), 991–1008.

M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996.


  • There are currently no refbacks.

Partnerzy platformy czasopism