Advanced Search
Article Contents
Article Contents

Existence of ground state solution and concentration of maxima for a class of indefinite variational problems

  • * Corresponding author

    * Corresponding author 

Claudianor O. Alves was partially supported by CNPq/Brazil 304804/2017-7

Abstract Full Text(HTML) Related Papers Cited by
  • In this paper we study the existence of ground state solution and concentration of maxima for a class of strongly indefinite problem like

    $ \begin{cases} -\Delta u+V(x)u = A(\epsilon x)f(u) \quad \mbox{in} \quad \mathbb{R}^{N}, \\ u\in H^{1}( \mathbb{R}^{N}), \end{cases} \qquad\qquad\qquad{(P)_{\epsilon}} $

    where $ N \geq 1 $, $ \epsilon $ is a positive parameter, $ f: \mathbb{R} \to \mathbb{R} $ is a continuous function with subcritical growth and $ V,A: \mathbb{R}^{N} \to \mathbb{R} $ are continuous functions verifying some technical conditions. Here $ V $ is a $ \mathbb{Z}^N $-periodic function, $ 0 \not\in \sigma(-\Delta + V) $, the spectrum of $ -\Delta +V $, and

    $ 0 < \inf\limits_{x \in \mathbb{R}^{N}}A(x)\leq \lim\limits_{|x|\rightarrow+\infty}A(x)<\sup\limits_{x \in \mathbb{R}^{N}}A(x). $

    Mathematics Subject Classification: Primary: 35B40, 35J20; Secondary: 47A10.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] C. O. Alves and G. F. Germano, Ground state solution for a class of indefinite variational problems with critical growth, J. Differ. Equ., 265 (2018), 444-477.  doi: 10.1016/j.jde.2018.02.039.
    [2] A. Ambrosetti and A. Malchiodi, Concentration phenomena for for NLS: recent results and new perspectives, in Perspectives in Nonlinear Partial Differential Equations, Contemporary Mathematics, Vol. 446, American Mathematical Society, Providence, RI, (2007), 19–30. doi: 10.1090/conm/446/08624.
    [3] A. AmbrosettiM. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations, Arch. Ration. Mech. Anal., 140 (1997), 285-300.  doi: 10.1007/s002050050067.
    [4] H. Brézis and E. H. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.  doi: 10.2307/2044999.
    [5] J. Chabrowski and A. Szulkin, On a semilinear Schrödinger equation with critical Sobolev exponent, Proc. Amer. Math. Soc., 130 (2001), 85-93.  doi: 10.1090/S0002-9939-01-06143-3.
    [6] M. del Pino and P. L. Felmer, Local mountain pass for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differ. Equ., 4 (1996), 121-137.  doi: 10.1007/BF01189950.
    [7] J. M. B. do Ó and B. Ruf, On a Schrödinger equation with periodic potential and critical growth in $\mathbb{R}^2$, Nonlinear Differ. Equ. Appl., 13 (2006), 167-192.  doi: 10.1007/s00030-005-0034-3.
    [8] A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal., 69 (1986), 397-408.  doi: 10.1016/0022-1236(86)90096-0.
    [9] M. F. Furtado and R. Marchi, Existence of solutions to asymptotically periodic Schrödinger equations, Electron. J. Differ. Equ., 2017 (2017), 1-7. 
    [10] W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to a semilinear Schrödinger equation, Adv. Differ. Equ., 3 (1998), 441-472. 
    [11] P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Part Ⅱ, Ann. Inst. Henri Poincare - Anal. Non Lineaire, 4 (1984), 223-283. 
    [12] G. B. Li and A. Szulkin, An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math., 4 (2002), 763-776.  doi: 10.1142/S0219199702000853.
    [13] Y. G. Oh, Existence of semi-classical bound states of nonlinear Schrödinger equations with potentials of the class $(V)_\alpha$, Commun. Partial Differ. Equ., 13 (1988), 1499-1519.  doi: 10.1080/03605308808820585.
    [14] Y. G. Oh, On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential, Commun. Math. Phys., 131 (1990), 223-253.  doi: 10.1007/BF02161413.
    [15] A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math., 73 (2005), 259-287.  doi: 10.1007/s00032-005-0047-8.
    [16] A. Pankov and K. Pflüger, On a semilinear Schrödinger equation with periodic potential, Nonlinear Anal. Theory Methods Appl., 33 (1998), 593-609.  doi: 10.1016/S0362-546X(97)00689-5.
    [17] P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, Reginal Conference Series in Mathematics, Vol. 65, American Mathematical Society, Providence, RI, 1986. doi: 10.1090/cbms/065.
    [18] P. H. Rabinowitz, A note on semilinear elliptic equation on $\mathbb{R}^N$, Nonlinear Analysis: A Tribute in Honour of G. Prodi, Quad. Scu. Norm. Super. Pisa., (1991), 307–318.
    [19] P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.  doi: 10.1007/BF00946631.
    [20] A. Szulkin and T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257 (2009), 3802-3822.  doi: 10.1016/j.jfa.2009.09.013.
    [21] A. Szulkin and T. Weth, The method of Nehari manifold, in Handbook of Nonconvex Analysis and Applications, International Press, (2010), 597–632.
    [22] M. Schechter, Linking Methods in Critical Point Theory, Birkhäuser, Boston, 1999. doi: 10.1007/978-1-4612-1596-7.
    [23] M. Schechter, Nonlinear Schrödinger operators with zero in the spectrum, Z. Angew. Math. Phys., 66 (2015), 2125-2141.  doi: 10.1007/s00033-015-0511-4.
    [24] M. Schechter and W. Zou, Weak linking theorems and Schrödinger equations with critical Sobolev exponent, ESAIM Control Optim. Calc. Var., 9 (2003), 601–619. (electronic) doi: 10.1051/cocv:2003029.
    [25] X. H. Tang, Non-Nehari manifold method for asymptotically periodic Schrödinger equations, Sci. China Math., 58 (2015), 715-728.  doi: 10.1007/s11425-014-4957-1.
    [26] X. H. Tang, New super-quadratic conditions for asymptotically periodic Schrödinger equation, Canadian Math. Bull., 60 (2017), 422-435.  doi: 10.4153/CMB-2016-090-2.
    [27] X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Commun. Math. Phys., 53 (1993), 229-244. 
    [28] M. Willem, Minimax Theorems, Birkhauser, 1996. doi: 10.1007/978-1-4612-4146-1.
    [29] M. Willem and W. Zou, On a Schrödinger equation with periodic potential and spectrum point zero, Indiana Univ. Math. J., 52 (2003), 109-132.  doi: 10.1512/iumj.2003.52.2273.
    [30] M. Yang, Ground state solutions for a periodic Schrödinger equation with superlinear nonlinearities, Nonlinear Anal., 72 (2010), 2620-2627.  doi: 10.1016/j.na.2009.11.009.
    [31] H. ZhangJ. Xu and F. Zhang, Ground state solutions asymptotically periodic Schrödinger equations with indefinite linear part, Math. Meth. Appl. Sci., 38 (2015), 113-122.  doi: 10.1002/mma.3054.
    [32] H. ZhangJ. Xu and F. Zhang, On a class of semilinear Schrödinger equation with indefinite linear part, J. Math. Anal. Appl., 414 (2014), 710-724.  doi: 10.1016/j.jmaa.2014.01.001.
  • 加载中

Article Metrics

HTML views(1494) PDF downloads(223) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint