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Existence and concentration of solution for a class of fractional elliptic equation in \(\mathbb {R}^N\) via penalization method

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Abstract

In this paper, we study the existence and concentration of positive solution for the following class of fractional elliptic equation

$$\begin{aligned} \epsilon ^{2s} (-\Delta )^{s}{u}+V(z)u=f(u) \quad \text{ in } \; \mathbb {R}^{N}, \end{aligned}$$

where \(\epsilon \) is a positive parameter, f is a continuous function having a subcritical growth, V is a continuous potential possessing a local minimum, \(N > 2s,\) \(s \in (0,1)\) and \( (-\Delta )^{s}u\) is the fractional laplacian.

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References

  1. Alves, C.O., do Ó, J.M.B., Souto, M.A.S.: Local mountain-pass for a class of elliptic problems involving critical growth. Nonlinear Anal. 46, 495–510 (2001)

  2. Alves, C.O., Figueiredo, G.M.: Multiplicity and concentration of positive solutions for a class of quasilinear problems via penalization methods. Adv. Nonlinear Stud. 11(2), 265–294 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alves, C.O., Souto, M.A.S.: On existence and concentration behavior of ground state solutions for a class of problems with critical growth. Commun. Pure Appl. Anal. 3, 417–431 (2002)

    MathSciNet  MATH  Google Scholar 

  4. Bartsch, T., Pankov, A., Wang, Z.-Q.: Nonlinear Schrödinger equations with steep potential well. Commun. Contemp. Math. 3, 1–21 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bartsch, T., Wang, Z.-Q.: Existence and multiplicity results for some superlinear elliptic problems on \(\mathbb{R}^N\). Commun. Partial Differ. Equ. 20, 1725–1741 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  6. Brändle, C., Colorado, E., Sánchez, U.: A concave-convex elliptic problem involving the fractional Laplacian. Proc. R. Soc. Edinb. A 143, 39–71 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cabré, X., Sire, Y.: Nonlinear equations for fractional laplacians, I: regularity, maximum principles, and Hamiltonian estimates. Ann. Inst. H. Poincaré Non Linéare 31, 23–53 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Caffarelli, L., Silvestre, L.: An extension problems related to the fractional Laplacian. Commun. Partial Differ. Equ. 32, 1245–1260 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chang, X., Wang, Z.Q.: Nodal and multiple solutions of nonlinear problems involving the fractional Laplacian. J. Differ. Equ. 256, 2965–2992 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chen, G., Zheng, Y.: Concentration phenomena for fractional noninear Schrödinger equations. Commun. Pure Appl. Anal. 13, 2359–2376 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dávila, J., del Pino, M., Wei, J.C.: Concentrating standing waves for the fractional nonlinear Schrödinger equation. J. Differ. Equ. 256, 858–892 (2014)

    Article  MATH  Google Scholar 

  12. del Pino, M., Felmer, P.L.: Local mountain pass for semilinear elliptic problems in unbounded domains. Calc. Var. Partial Differ. Equ. 4, 121–137 (1996)

    Article  MATH  Google Scholar 

  13. del Pino, M., Felmer, P.L., Miyagaki, O.H.: Existence of positive bound states of nonlinear Schrödinger equations with saddle-like potential. Nonlinear Anal. 34, 979–989 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  14. Dipierro, S., Palatucci, G., Valdinoci, E.: Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian. Matematiche 68, 1 (2013)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. do Ó, J.M.B., Souto, M.A.S.: On a class of nonlinear Schrödinger equations in \(\mathbb{R}^{2}\) involving critical growth. J. Differ. Equ. 174, 289–311 (2001)

  17. Fall, M.M., Mahmoudi, F., Valdinoci, E.: Ground states and concentration phenomena for the fractional Schrödinger equation. arXiv:1411.0576v2 [math.AP] (2015)

  18. Felmer, P., Quass, A., Tan, J.: Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian. Proc. R. Soc. Edinb. A 142, 1237–1262 (2012)

    Article  MATH  Google Scholar 

  19. Floer, A., Weinstein, A.: Nonspreading wave packets for the cubic Schrödinger equations with bounded potential. J. Funct. Anal. 69, 397–408 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  20. Oh, Y.J.: Existence of semi-classical bound states of nonlinear Schrödinger equations with potentials on the class \((V)_a\). Commun. Partial Differ. Equ. 13, 1499–1519 (1988)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  22. Secchi, S.: Ground state solutions for nonlinear fractional Schrödinger equations in \(\mathbb{R}^N\). J. Math. Phys. 54, 031501–17 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  23. Shang, X., Zhang, J.: Concentrating solutions of nonlinear fractional Schrödinger equation with potentials. J. Differ. Equ. 258, 1106–1128 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  24. Shang, X., Zhang, J., Yang, Y.: On fractional Schrödinger equation in \(\mathbb{R}^N\) with critical growth. J. Math. Phys. 54, 121502–19 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Wang, X.: On concentration of positive bound states of nonlinear Schrödinger equations. Commun. Math. Phys. 53, 229–244 (1993)

    Article  MATH  Google Scholar 

  26. Zhang, J., Liu, X., Jiao, H.: Multiplicity of positive solutions for a fractional Laplacian equations involving critical nonlinearity. arXiv:1502.02222 [math.AP]

  27. Willem, M.: Minimax Theorems. Birkhäuser, Boston (1996)

    Book  MATH  Google Scholar 

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Acknowledgments

This paper was completed while the second author named was visiting the Department of Mathematics of the Rutgers University, whose hospitality he gratefully acknowledges. He would like to express his gratitude to Professor Haim Brezis and Professor Yan Yan Li for invitation and friendship. The authors want to thank the anonymous Referee for his or her deep observations, careful reading and suggestions, which enabled us to improve this version of the manuscript.

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Correspondence to Olímpio H. Miyagaki.

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Communicated by P. Rabinowitz.

Research of C. O. Alves partially supported by CNPq/Brazil Proc. 304036/2013-7 and INCTMAT/CNPq/Brazi. O.H.M. was partially supported by INCTMAT/CNPq/Brazil, CNPq/Brazil Proc. 304014/2014-9 and CAPES/Brazil Proc. 2531/14-3.

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Alves, C.O., Miyagaki, O.H. Existence and concentration of solution for a class of fractional elliptic equation in \(\mathbb {R}^N\) via penalization method. Calc. Var. 55, 47 (2016). https://doi.org/10.1007/s00526-016-0983-x

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