Home Nontrivial solutions of superlinear nonlocal problems
Article
Licensed
Unlicensed Requires Authentication

Nontrivial solutions of superlinear nonlocal problems

  • Giovanni Molica Bisci EMAIL logo , Dušan Repovš and Raffaella Servadei
Published/Copyright: May 1, 2016

Abstract

We study the question of the existence of infinitely many weak solutions for nonlocal equations of fractional Laplacian type with homogeneous Dirichlet boundary data, in presence of a superlinear term. Starting from the well-known Ambrosetti–Rabinowitz condition, we consider different growth assumptions on the nonlinearity, all of superlinear type. We obtain three different existence results in this setting by using the Fountain Theorem, which extend some classical results for semilinear Laplacian equations to the nonlocal fractional setting.


Communicated by Frank Duzaar


Funding statement: The authors were supported by the SRA Program P1-0292-0101 Topology, Geometry and Nonlinear Analysis. The first and the third author were supported by the INdAM-GNAMPA Project 2015 Modelli ed equazioni non-locali di tipo frazionario. The third author was supported by the MIUR National Research Project Variational and Topological Methods in the Study of Nonlinear Phenomena and by FP7-IDEAS-ERC Starting Grant 2011 #277749 EPSILON (Elliptic Pde’s and Symmetry of Interfaces and Layers for Odd Nonlinearities), founded by the European Research Council (ERC).

Acknowledgements

The authors warmly thank the anonymous referee for her/his useful and nice comments on the paper.

References

[1] Adams R. A., Sobolev Spaces, Academic Press, New York, 1975. Search in Google Scholar

[2] Alves C. O. and Liu S. B., On superlinear p(x)-Laplacian equations in N, Nonlinear Anal. 73 (2010), no. 8, 2566–2579. 10.1016/j.na.2010.06.033Search in Google Scholar

[3] Ambrosetti A. and Rabinowitz P., Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349–381. 10.1016/0022-1236(73)90051-7Search in Google Scholar

[4] Bartsch T., Infinitely many solutions of a symmetric Dirichlet problem, Nonlinear Anal. 20 (1993), no. 1, 1205–1216.10.1016/0362-546X(93)90151-HSearch in Google Scholar

[5] Binlin Z., Molica Bisci G. and Servadei R., Superlinear nonlocal fractional problems with infinitely many solutions, Nonlinearity 28 (2015), 2247–2264. 10.1088/0951-7715/28/7/2247Search in Google Scholar

[6] Brezis H., Analyse Fonctionelle. Théorie et Applications, Masson, Paris, 1983. Search in Google Scholar

[7] Cerami G., An existence criterion for the critical points on unbounded manifolds (in Italian), Ist. Lombardo Accad. Sci. Lett. Rend. Sez. A 112 (1978), no. 2, 332–336. Search in Google Scholar

[8] Cerami G., On the existence of eigenvalues for a nonlinear boundary value problem (in Italian), Ann. Mat. Pura Appl. (4) 124 (1980), 161–179. 10.1007/BF01795391Search in Google Scholar

[9] Costa D. G. and Magalhães C. A., Variational elliptic problems which are nonquadratic at infinity, Nonlinear Anal. 23 (1994), 1401–1412. 10.1016/0362-546X(94)90135-XSearch in Google Scholar

[10] Di Castro A., Kuusi T. and Palatucci G., Nonlocal Harnack inequalities, J. Funct. Anal. 267 (2014), 1807–1836. 10.1016/j.jfa.2014.05.023Search in Google Scholar

[11] Di Castro A., Kuusi T. and Palatucci G., Local behavior of fractional p-minimizers, Ann. Inst. H. Poincaré Anal. Non Linéaire 267 (2015), 1807–1836. 10.1016/j.anihpc.2015.04.003Search in Google Scholar

[12] Di Nezza E., Palatucci G. and Valdinoci E., Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), no. 5, 521–573. 10.1016/j.bulsci.2011.12.004Search in Google Scholar

[13] Fang F. and Liu S. B., Nontrivial solutions of superlinear p-Laplacian equations, J. Math. Anal. Appl. 351 (2009), no. 1, 138–146. 10.1016/j.jmaa.2008.09.064Search in Google Scholar

[14] Ferrara M., Molica Bisci G. and Zhang B., Existence of weak solutions for non-local fractional problems via Morse theory, Discrete Contin. Dyn. Syst. Ser. B 19 (2014), 2493–2499. 10.3934/dcdsb.2014.19.2483Search in Google Scholar

[15] Franzina G. and Palatucci G., Fractional p-eigenvalues, Riv. Math. Univ. Parma (N.S.) 5 (2014), no. 2, 373–386. Search in Google Scholar

[16] Jeanjean L., On the existence of bounded Palais–Smale sequences and application to a Landesman–Lazer type problem set on N, Proc. Roy. Soc. Edinburgh Sect. A 129 (1999), no. 4, 787–809. 10.1017/S0308210500013147Search in Google Scholar

[17] Liu S. B., On ground states of superlinear p-Laplacian equations in N, J. Math. Anal. Appl. 361 (2010), no. 81, 48–58. 10.1016/j.jmaa.2009.09.016Search in Google Scholar

[18] Liu S. B., On superlinear problems without Ambrosetti–Rabinowitz condition, Nonlinear Anal. 73 (2010), no. 3, 788–795. 10.1016/j.na.2010.04.016Search in Google Scholar

[19] Liu S. B. and Liu S. J., Infinitely many solutions for a superlinear elliptic equation, Acta Math. Sinica (Chin. Ser.) 46 (2003), no. 4, 625–630. Search in Google Scholar

[20] Miyagaki O. and Souto M., Superlinear problems without Ambrosetti and Rabinowitz growth condition, J. Differential Equations 245 (2008), no. 12, 3628–3638. 10.1016/j.jde.2008.02.035Search in Google Scholar

[21] Rabinowitz P. H., Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math. 65, American Mathematical Society, Providence, 1986. 10.1090/cbms/065Search in Google Scholar

[22] Schechter M. and Zou W., Superlinear problems, Pacific J. Math. 214 (2004), no. 1, 145–160. 10.1007/978-0-8176-4902-9_9Search in Google Scholar

[23] Servadei R., Infinitely many solutions for fractional Laplace equations with subcritical nonlinearity, Contemp. Math. 595 (2013), 317–340. 10.1090/conm/595/11809Search in Google Scholar

[24] Servadei R., The Yamabe equation in a non-local setting, Adv. Nonlinear Anal. 2 (2013), 235–270. 10.1515/anona-2013-0008Search in Google Scholar

[25] Servadei R. and Valdinoci E., Mountain Pass solutions for non-local elliptic operators, J. Math. Anal. Appl. 389 (2012), 887–898. 10.1016/j.jmaa.2011.12.032Search in Google Scholar

[26] Servadei R. and Valdinoci E., A Brezis–Nirenberg result for non-local critical equations in low dimension, Commun. Pure Appl. Anal. 12 (2013), no. 6, 2445–2464. 10.3934/cpaa.2013.12.2445Search in Google Scholar

[27] Servadei R. and Valdinoci E., Lewy–Stampacchia type estimates for variational inequalities driven by (non)local operators, Rev. Mat. Iberoam. 29 (2013), no. 3, 1091–1126. 10.4171/RMI/750Search in Google Scholar

[28] Servadei R. and Valdinoci E., Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst. 33 (2013), no. 5, 2105–2137. 10.3934/dcds.2013.33.2105Search in Google Scholar

[29] Servadei R. and Valdinoci E., The Brezis–Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc. 367 (2015), no. 1, 67–102. 10.1090/S0002-9947-2014-05884-4Search in Google Scholar

[30] Struwe M., Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Ergeb. Math. Grenzgeb. (3), Springer, Berlin, 1990. Search in Google Scholar

[31] Willem M., Minimax Theorems, Birkhäuser, Basel, 1996. 10.1007/978-1-4612-4146-1Search in Google Scholar

[32] Willem M. and Zou W., On a Schrödinger equation with periodic potential and spectrum point zero, Indiana Univ. Math. J. 52 (2003), 109–132. 10.1512/iumj.2003.52.2273Search in Google Scholar

[33] Zou W. M., Variant fountain theorems and their applications, Manuscripta Math. 104 (2001), 343–358. 10.1007/s002290170032Search in Google Scholar

Received: 2015-10-13
Revised: 2016-1-4
Published Online: 2016-5-1
Published in Print: 2016-11-1

© 2016 by De Gruyter

Downloaded on 24.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/forum-2015-0204/html
Scroll to top button